Mostly look for dents in the frame or serious rust on frame or other components. Don't worry about rust on a chain, though, chains are easily replaced, and they may clean up and work fine after being lubed.

Try to get alloy wheels rather than steel wheels - take a magnet to check if you're not sure.

Check that the wheels spin smoothly without any grinding sound in the bearings, ditto for the cranks (bottom bracket). Other than that, just check that the shift levers move smoothly. It's pretty common for an old bike to need a little cleaning, oiling and adjusting, so don't worry too much if everything isn't perfect, just make sure it appears to work. The best buys may have been sitting in a garage or basement for years and just need a little attention get back on the road.

Oh, and I wouldn't worry too much about a specific brand, frame tubing or component level. Even low level Suntour or Shimano shifters and derailers will work fine for basic transportation. In fact I'd tend to avoid exotic and collectable European bikes as replacement parts would likely be more expensive and/or harder to find.