Bike fit isn't exactly rocket science. It doesn't really require expensive equipment and it doesn't have to be done by experts. Sure, it's cool to pay someone to fit you for a bike, but it's not that necessary. It makes more sense if you are into competitive cycling and are buying a whole new bike and every ounce of performance matters. However, to get you fit to your existing bike all you can do is to move things around, maybe replace a stem and most people should be able to do that themselves. That cash would be better spent on a quality saddle or new grips or bar tape.

Your bike doesn't need to be the perfect size either, as long as it's not drastically too small or too large you can compensate for incorrect bike size by changing the stem, the handlebars or the seat post.

Before you start troubleshooting: about weight distribution and bike fit
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Before you start troubleshooting your bicycle related aches and pains it would help to understand the weight distribution. Your body touches your bike (interfaces with) in three points: your seat, your feet and your hands. Your body weight is distributed between these three points and if the weight isn’t distributed in a way that your body can tolerate it one of those three areas will start causing problems that may affect various parts of your body and cause discomfort and pain and make you miserable. This, of course, will vary from person to person.

It’s obvious how you support your weight with your behind and your arms on a bike: the first rests on the saddle the second rest on the handlebar. However, people are often confused about how it is that your supporting your weight with legs if your legs are moving in circles and you’re not standing on the ground? That’s because when you pedal you apply downward force to the pedals which in turn creates a counter force pointing upwards which lifts your upper body with every pedal down-stroke, therefore supporting part of your weight. Notice, how you suddenly feel the saddle pressure more when you’re coasting. That’s because you’re not applying any force with your legs.

Your weight distribution also varies during a bike ride.
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It’s not always the same. When you coast downhill most of your weight is probably supported by your seat. When you climb up a steep hill almost all your weight is carried by your legs, you’re actually pulling the handlebar, not resting on it. Even more so if you’re standing on the pedals. When you pedal along a flat road your weight is distributed more or less evenly between all the three points.

What really matters for most people is the total of time you spend sitting on your saddle and how the weight is distributed during your time on the saddle.

There is no single, universal approach to bike rider weight distribution and to bike fit because that depends on many things: your body type and size, your physical fitness, your body response (different people can develop different problems in response to the same trigger), the type of your bike and your ridding style and needs. There are some aspects of bike fit that are known to be necessary for comfortable fit, universal for all body types and most people agree on, such as the saddle height, but there are some that are being argued and depend on your physical characteristics, such as the forward body angle.

Also, bike fit isn’t all that important for short rides, like an average commute, or nobody would be riding City Bikes ;) In the simplest terms, a bike can be fit either for performance or for comfort or somewhere in between, but you can’t have extreme performance and extreme comfort at the same time, it’s always a compromise. There is also the ugly reality that a 45 years old person with 15 extra pounds around the waist just can’t be set up on a bike the same way a 25 years old, fit person can. What needs to ultimately happen is that you have to become comfortable on your bike. If you can enjoy the ride without pain and numbness and have fun on the saddle then you’re fit properly no matter what people say.

Do not be obsessed with bike fit
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What I'm trying to say is don't be obsessed with bike fit. You can be fit by a pro and still be miserable on the bike or you can be comfortable but your fit is off according to most fit standards.

I suggest you search Google for bike fit advice, there are plenty of instructions so pick the one that makes most sense to you and then, if you experience any of the issues I describe here, make the necessary adjustments. I personally find this video from Performance Bicycle to be a no-nonse guide to road bike fitting:

https://www.youtube.com/watch?v=FAl_5e7bIHk

You can even do it without a trainer if you have a friend who can watch you as you ride around.

Most common problems people experience and some tips how to address them
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Remember that some of the symptoms may appear together. Make adjustments in small increments. Write down what you have adjusted and by how much. Make one adjustment at a time and test the results. If it doesn’t help, undo what you’ve done and try something else before mixing different solutions. Take your time, even if that means few days.

Side note about stems
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A lot of adjustments can be made by manipulating the stem and it should be the first thing you try if you have problems with arms, neck and back. This is because the saddle position is relative to the pedals and once you get the correct saddle and leg position for comfortable and effective pedaling you won’t really be able to (and shouldn’t) adjust the saddle further (other than its angle) to address issues with your arms, neck and back. If you do and mess up the legs position, you’ll be back at the beginning and the whole fit will be off.

That’s where the stem comes in. A stem allows you to move the handlebar up and down, forward and backward and compensate for the frame size without messing up your leg position too much. If you’re really trying to nail it down get an adjustable stem, there are ones where you can adjust the angle, the length or both. They’re unsightly but they’re great for troubleshooting and fitting. Once you have your position dialed in, you can replace it with a fixed stem that matches the current length and angle of your adjustable stem.

Side note about saddles
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Many people make the newbie mistake of buying large, cushioned saddles. That doesn't help. After 30 miles on the saddle it won't matter how cushioned your saddle is. Even the softest saddle will feel hard after some time on it. Some cushioning is good, of course, but in the long term it's your behind than needs to be fixed and that is only done by spending time on the saddle. The truth is that no matter what you do your behind will hurt in the beginning. With time, it'll become tougher and your pains will go away.

What is more important than cushioning is the saddle shape and the way it's set up. Your saddle should be the correct width which is pretty close for most people, but some will do better with more narrow or wider saddle than others. Your two sitting bones should rest comfortably on the wider part of the saddle and you shouldn't be sliding forwards or backwards on the saddle.
Now, on to the common problems.

Symptom: your seat hurts
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Possible causes: You’re putting too much weight on your rear and not enough on your arms and legs and that’s because your stem may be too short or too high up (handlebars too high up), your saddle may be too far back in relation to pedals; your saddle may also be way too low so you’re not supporting enough of your weight with your legs; or you simply didn’t have enough time on the saddle, as mentioned above it’s very common to experience pain in the rear if you haven’t ridden for a long time, also your saddle may be too wide or too narrow.

Solutions: make sure that your saddle is level at first and then play with the angle; move it slightly forward, adjust the saddle to the correct height, get a longer stem to put more weight on your hands, lower the handlebar, try a different saddle, perhaps a more narrow one; try riding more until your behind gets used to this.

Symptom: kind of like the above but your main problem is that your crotch skin seems to be on fire
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Possible causes: your saddle is too high up and you rock sideways rubbing your skin; you frequently slide forward and backward on your saddle; you wear something that has seams in that area or is made of rough fabric; you have a cheap saddle; you sweat a whole lot and the sweat irritates your skin; you have some medical condition.

Solutions: set the saddle height and fore-aft position properly so you stay put on the saddle, get a better saddle; wear better clothing that has no seams, that wicks your moisture and that offers enough ventilation; keep that area clean, use Butt’R lubricant on long rides, check with a doctor if the rash persists.

Symptom: your hands hurt (wrists or elbows)
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Possible causes: you’re putting too much weight on your hands because your stem may be too long or too low (handlebars too low); your saddle may be too far forward or tilted down (nose down); your wrists and palms receive too much road vibration; you stiffen up elbow joints or "death-grip" the bar or your hands are simply weak.

Solutions: make sure your saddle is level, move your saddle backwards a bit, get a shorter stem or one that is more angled upwards, try to raise your handlebar; if it’s mostly your palms and wrists that hurt then try padded gloves, gel handlebar tape or gel grips or lower your front tire pressure a little bit, relax your hands, allow your joints to be bent, if you have an aluminum bike consider steel or carbon fork, strengthen your arms through weight exercise.

Symptom: your neck hurts
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Possible causes: your stem/handlebar is too low so you’re bent down too much and constantly hold you head up to see the road; your back and neck muscles are weak or stiff; you have a big belly that pulls your core down and strains your back and neck; you’re a woman with large breasts.

Solutions: Raise the stem or replace it with one that is angled up more and/or shorter, raise your handlebar; work out your back muscles and stretch them regularly, do some yoga, lose weight. If you’re a woman with large breasts wear tight, supportive sport bra to prevent your breasts from pulling down. You may be able to lower the handlebar back again once your neck and back get stronger and more flexible.

Symptom: you get a headache
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Possible causes: same as “your neck hurts” above: neck discomfort can cause tension headaches.

Solutions: same as “your neck hurts” above.

Symptom: your lower back hurts
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Possible causes: you’re either stretched too much to reach the handlebar; or your saddle is to high up and you move your pelvic area sideways too much when pedaling (these results in lower back muscle pain) or, the opposite: your position is too upright, your spine is too straight and it isn’t absorbing shocks from road bumps properly (this results in pain around lower spine mainly, not really the muscles).

Solutions: raise the stem and the handlebar, lower your saddle (although remember: that can compromise your leg position) to bring your torso up but avoid being completely upright. Your spine acts as a spring and it needs to be bent, arched so it can flex (like a bow) and absorb and dissipate shocks. If your spine is straight up, it compresses vertically instead of flexing and that is very unhealthy and can lead to back problems, popped disks, pinched nerves and all sort of nasty things. If you can’t rise the handlebar enough and have to lower the saddle significantly then you need a new bike. Riding with saddle too low can and will result in knee pain. However, if the pain persists you should probably consult a doctor.

Symptom: you slide forward on your seat
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Possible causes: you’re too far from the handlebar so you’re trying to pull yourself towards the handlebar and that may be because: the bike is too big for you, the stem is too long, the saddle is too far back or you’re sliding because the saddle’s nose is tilted down.

Solutions: first make sure the saddle is level or nose tilted up slightly then experiment with the angle, try shorter stem, slide your saddle forward, get a smaller bike.

Symptom: you slide backwards on your seat
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Possible causes: you’re too close to the handlebar so you’re trying to push yourself away from the handlebar and that may be because: the bike is too small for you, the stem is too short, the saddle is too far forward, the saddle is too low or or you’re sliding because the saddle’s nose is tilted up.

Solutions: make sure the saddle is level then experiment with the angle, set the correct height for your saddle, slide saddle backwards, get longer stem, get a larger bike.

Symptom: your knee(s) hurt on the front, below the kneecap, or inside the knee (the joint)
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Possible causes: you’re straining your knees by mashing the pedals and not letting your legs expand for a full stroke because your saddle is too low or too far forward; your cleats are not placed properly for optimal leg position, or the cleats don’t have enough float to allow your knees to twist naturally and relax; you may also have a medical condition.

Solutions: raise your seat and/or move it slightly backwards, adjust the cleats so that the center of the pedal axle is aligned with center of the ball of your foot; if you can, adjust the pedal/cleat float so you can wiggle your feet without unclipping to allow your knees to twist and flex, your knees need to be able to flex sideways during movement or they will get stiff and start to hurt. If nothing helps, check with a doctor because if this goes on for too long you can ruin your knees for life. NOTE: some cleats come in "no float" version so you should replace them with ones that have few degrees of float.

Symptom: your leg(s) hurt behind the knee, on the “inside”, the soft parts on the back of the knee
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Possible causes: your saddle is too high up and you’re over-extending your legs and strain your tendons, your cleats are not placed properly, your shorts’ elastic hem is too tight.

Solutions: lower your saddle, adjust the cleats so that the center of the pedal axle is aligned with center of the ball of your foot, wear shorts with more relaxed elastic hem to avoid excessive squeeze, it may seem like not much but it all adds up on long rides.

Symptom: your private parts get numb
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Disclaimer: I'm a guy so I don't know how this affects women, but from what I heard this can affect women too, but dudes suffer from this more often.

Possible causes: Too much pressure on the perineum area so tissue gets compressed and cuts blood flow to your private parts, clothes too tight, clothes not warm enough, a medical condition.

Solutions: get a better saddle with a cutout or a channel and quality gel padding, set the saddle height and angle correctly, wear decent cycling shorts that are not too tight, in cold weather wear extra clothes, warm underwear or stuff a wool sock in there, check with your doctor if the problem won’t go away. If this goes on for too long you may end up with erectile dysfunction syndrome. A leather Brooks saddle could be a good option but it takes a very long time to break in before it becomes truly comfortable. Often hundreds of miles. But once worn in it is pretty much the only type of a saddle that can be ridden on for extended periods of time without padded chamois.