Ah, the Varsity. My first foray into riding classic/vintage bikes was one of those, and I have the frame stripped down sitting waiting for something to be done with it.
It's not unreasonable to strip everything off and replace only the front brake and wheelset (and chain, probably). A decent starting wheelset can be had for $100-$130 if you look around at Velomine, eBay, eighthinch, etc. Replacing the one piece crankset and bottom bracket is a good idea, and will run about $100 more, but it isn't necessary and you can definitely use the stock crankset/BB.
If you want to get into DIY/bike building, it's not a terrible idea to buy a good starting wheelset, slap it on the Varsity frame, and then save up for a better frame/parts that you can transfer the wheels too later. I've seen a good amount of bikes on Craigslist that were good candidates for conversions that happened to be missing only a wheelset.
Buying from Bikesdirect is the other option, it really depends on if you want to get into the nitty-gritty of bike building (you will have to put a pre-built bike together anyway though, it's just not as technical).