It depends what you want and how much time and money you want to spend. If you're looking for a ground up, show-worthy restoration, then my advice would be to stick to something that already has a big following and good aftermarket support. A lot of your 1960s muscle cars will have clubs and retro parts already. These include your Fords and Chevy's. At the moment, one of the most popular classic cars to fix up, and one that has massive aftermarket support, is the Ford Mustang, especially the years from 1964 1/2 to '68, with lots of great parts from the later years available as well. When these cars first came out they sold like crazy, so there's a lot of them out there, and parts, both original and reproduction, are easy to find. They would make a great first project.

How much do you want to do? If this is your first time fixing up a classic car, you should look for one that is mechanically sound, unless you plan to rebuild the engine or replace it. If you have one that's drivable, then you can drive and fix it up at the same time. I'd go with either the older sports cars, or the later American cars, ones that parts can still be found for and can be resold for a little profit. Make friends with people who know how to fix up old cars, and join clubs for you make and model. Make sure you have a place to work on the car, and invest in some good, sturdy tools as well, you'll need them. Oh, and learning to weld might be a good idea as well.