You can usually ride for a long time (years?) on a bearing that's in less than great shape, though depending on the type of bearing it might be a good idea to fix it so you don't do more damage. $29 is a good price to fix a bearing since they can be labor intensive.
There are four (or five) major bearings that I can think of: Each hub, the bottom bracket (where the crankset spins), the headset (where the handlebars turn), and maybe the freehub (ratchet for the rear) depending on the type. A lot of manufacturers have moved to sealed cartridge bearings for new and high end parts, which means that once it wears out you throw it out and buy a new one. The advantage is better tolerances and lower friction, but the downside is that they are not really serviceable. At least cartridge bearings are usually not too expensive. Older bikes like yours will likely have cup and cone bearings for most of the major bearings (I don't know much about older bottom brackets though). They usually have some way to adjust the preload pressure on the bearing, and the loose balls are generally replaceable. I can't think of any way riding a cartridge bearing until it dies would damage anything, but if you neglect cup and cone bearings you can do some damage to the races. If one of the races is the hub, which is laced into the wheel, that can get to be a difficult or expensive repair.
You can tell if something is not right in a bearing if it is rough when it spins, if it binds, if it's loose, or if it makes excessive noise. Sometimes you can fix by taking it apart, cleaning it, and lubing it, and readjusting it when you put it back together. Otherwise you might need to replace parts. If you take it apart, you will want to look at the condition of the races that the balls run in. They should be smooth when checked with a ballpoint pen.
To take apart hubs, you usually just need a couple cone wrenches-- special thin open end wrenches that allow you to adjust the coned. You will need two in the size that your hub uses, and they should not be used for anything else. You will also need an adjustable wrench or appropriate size metric open end wrench for the lock nut outside the cone. I won't go into detail, but you basically adjust the hub until it's tight enough that there's only a tiny amount of play which goes away when you clamp the wheel in the bike.
Threadless headsets are easy to adjust, you just need a couple allen wrenches. You can take it apart, clean it (really well. Any tiny dirt in the bearings will cause them to wear out very quickly), lube it, and put it back together. Replacing a headset is an entirely different task that requires specialized tools and has some risk of ruining your frame if done wrong.
Threaded bottom brackets are easy enough to deal with once you have the correct tools. You will need a crank arm puller to take the crank arms off, and you will need a bottom bracket tool to fit the splines on your bottom bracket (maybe $5-10). You will also need a really big wrench (a foot long or more is preferable). Standard threaded bottom brackets with square taper are inexpensive and you can usually just replace the whole thing.
I almost forgot that there are also bearings in the pedals. They usually go together like hubs, but it can be difficult to get a wrench or socket in there. Low-end pedals are meant to be disposable, I think.
If you want to learn more, I would pick up a good maintenance book like "The Zinn and Art of Mountain Bike Maintenance" (there's a road version too). I can also answer questions relating to types of components I've worked on. I used to work as a volunteer maintaining a small fleet of rental bikes so I've had a chance to try almost every kind of repair or maintenance you can do on modern low-end mountain bikes.
I like learning bike maintenance because I like understanding how things work and fixing them myself rather than relying on others. It might take some time to learn, but with practice you can do simple stuff quickly and save time and money by not going to bike shops. You will also be able to recognize when something is beyond your ability to fix and then take it to a bike shop. You will also figure out what tools are worth buying and which ones are not cost effective to own.
