A bare frame costs more than $125 most of the time. With a more expensive bike, I'd tell you to watch for drivetrain wear. That can be quite expensive to fix. But it's hard to go wrong at this price - even if you have to replace all the wear parts, you'll still be doing okay. And odds are on your side...
Here's a short article on the fastest part of drivetrain wear.
Chain Maintenance
Chains and gears wear sympathetically. If someone trashes a chain and leaves it on, shifting may still feel okay. But sooner or later the chain will break and a new chain will have really horrible performance on the existing gears, requiring you to replace a bunch of stuff. If a bike shifts well and the chain has an acceptable amount of stretch, you're good. If it shifts poorly and the chain has stretch, you won't know until you put a new chain on. If it shifts well and has a lot of stretch, again, you won't know until you put the new chain on... but it's a worrisome sign. So you might take a steel tape measure with you and see what's up.
If you squeeze the brakes and try to rock the bike fore and aft, you'll probably feel some flex. But nothing should go "clunk." For a front brake, that can be a bad headset, bad fork or bad hub. Rear brake, probably the hub. Some brakes develop a little play over time too, though.
