Chain rings aren't too expensive, and there is no need to buy anything fancy. If you don't use the granny gear (the smallest ring up front)very often, you may not need to change that. Usually the bigger two are more used and worn out. Do you know what brand your cranks are? You can always buy direct replacements, or you could look on a website like Jenson USA or Performance. You'll want to look at a rings for a "triple crank setup" (i.e. you don't want 1X or single speed rings). The only other thing you really need to know is how many teeth are on each ring.
If your current cassette is an 8-speed shimano, any 8 speed shimano will work. Look for Shimano Deore or SLX level components. They should be fairly cheap.
With your chain, make sure you get the right one (8 speed, 9 speed, 10 speed, etc.). And if you don't know the method for determining the proper chain length when you have to put on the new chain, you can count the links on the old one (assuming that one works fine and was installed properly). You will need a chain breaker. You can also buy a removable master link, they are easier to use than trying to get a chain link pin installed properly. Once installed, a master link will allow you to break your chain without needing a special tool.
Sounds like you're on the right track. And you're right, most of it isn't that hard to do. Especially once you've done it once or twice.