I’ve got a 1998 Buick Century Custom and the serpentine belt is beginning to fray. Today I opened the hood and there was a 1mm strip of the belt just laying on the engine block. I think it might be time to replace it 
What do I need in order to do this on my own; or, am I better off buying a belt and taking it to a mechanic?
are you at least a little bit mechanically adept?
if you are a serpentine belt change is really easy.
a word of warning though: make sure the keys are NOT in the car when doing this, so no one accidentally starts the car while you have your fingers under the hood. AND if you get the old one off, and cant get the new belt on correctly you WONT be able to drive it to a mechanic to finish the job.
this would cost around 80 to 100 bucks to have done, so you have to measure your aggravation and the pinched fingers to see if it is worth it to save the $$. a belt (you doing it) should cost around 15 to 25 bucks( http://www.autozone.com/N,15300077//shopping/partTypeResultSet.htm . so the extra savings are worth it, if you dont get frustrated, and have the time and patience.
the most important thing to take note of is to make a diagram, take a picture (s) of the current path of the old belt. they only go on one way, and you must put the new one on the same route.
there is a movable tensioner pulley in the system. with the hood up, and while pushing down on the belt, look for the pulley which moves (not rotates, but the whole pulley moves)
looking at the tensioner on the arm which mounts it to the block. there should be a 3/8" square hole in it. (it MAY be a 1/2" though) (some vehicles have, instead of a square hole, a cast square part on the arm, which lets you use a wrench instead of a ratchet.) you use either a ratchet or a wrench as appropriate. (either with or without a short stubby extension as needed) to turn the pulley. you will notice when you turn the pulley the belt gets loose.
when the belt is loose you can wiggle the belt off. sometimes you have to work to find the easiest pulley to work from. (the smaller hub pulleys are usually the best place to start. you will notice things like the AC pump is a large pulley with a big lip thats a #$%^ to start with)
sometimes this is a two person job, one to hold the ratchet/wrench to keep the belt loose, and one to move the belt.
once you get the belt off the first pulley stop and REALLY carefully look with a flashlight to see exactly how the belt goes around all the pulleys. note that some pulleys the belt is not riding on the grooves, but on the flat side. make sure the new one goes on the same way.