Congrats!

Find the Serial Number on the bottom bracket shell (where the crank axle passes through the frame) and check here for the exact year:

http://www.vintage-trek.com/SerialNumbers.htm

If it appears in ride ready condition, I would simply ride it. Don't worry about finding the problems, the problems will find you.

However, to be on the safe side, you might want to have it looked over by a trusted bike shop or friend. At minimum you want to make sure your braking system (levers, cables and pads) is in optimal condition, and that your tires won't blow out on you.

Eventually (maybe sooner) you'll want to overhaul all bearing assemblies, lube dérailleur pivots points, and lube the chain.