My first fixie was an old schwinn - continental which is like the varsity - and I would advise against it. They have a single piece crank and most of them have odd sized parts so finding good new parts can be tricky. Of the stuff you are likely to find at the thrift store, look out for Univega, Japanese made Schwinns from the 80s (Travellers, Le Tours ...) Fuji, Nishiki ... really, anything lugged with horizontal drop-outs and three piece cranks made in the mid-late 70s or 80s should work fine.

For me, it is worth droping some cash on a new wheel set (search for IRO wheelsets on this forum) so don't worry about the wheels on the bikes you find. You may also find that a new set of handle bars and a new seat will make good investments (but I think this is less of a worry than the wheels). But when you're looking at used bikes, try to figure out if the stem, seat post, headset, bottom bracket and cranks are going to be usable. Individually, these things don't cost that much but if you need all of them you will rack up quite a bill.

If you haven't done so already, spend lots of time at fixedgeargallery.com and see what other folks have done. There are loads of conversion bikes there and they will give you ideas about what to look for.

Good luck and let us know what you come up with.