CASSETTE, CHAIN, AND CHAINRINGS
First, inspect the cassette and chain. Look to make sure they are clean and show no rust or excess buildup of chain lube, and that they are not overly dry and squeaky as you back-pedal the cranks.
Then look closer at the drivetrain to assess the wear on it. You are looking specifically for wear on the trailing edge of the cassette teeth, which you can find by running your fingernail over the edge. If there is a burr, the drivetrain is basically at the end of its usable life. It may work fine today, but it will likely need to be replaced soon. With a burr present on even just one tooth of the cassette, the cassette cannot take a new chain without skipping and running poorly. If there are burrs on most cassette teeth, it likely means that the chain is stretched to the point of damaging the chain rings.
Now look at the inside of the chainrings for scarring and wear. If there is significant wear on the inside, and the cassette also shows wear, know that you are in for a rather expensive drivetrain repair if you purchase the bike.
Some drivetrain wear is to be expected with any used bike, but if the drivetrain (which is about the easiest thing on a bike to maintain) seems neglected or in ill repair, run—don’t walk—away from the bike, as it is likely that other items that are more subtle will also be an issue.
Proper maintenance of a drivetrain is simple: Apply a few drops of chain lube to each link of the chain and to the bearings or bushings of the rear derailleur, then towel clean. The aim is to clean the drivetrain of contaminants, not soak it—dust and mud will collect on the wet spots.
Part and Labor Estimates:
New Cassette: $40-$100
New Chain: $30-$50
New Chainrings: $30 and up, each
New Drivetrain Install (labor): $50 and up
NEXT: DISC BRAKES