Cool! Do you think it would be better to just get a Pista and get better parts as time goes on?
Personally, I don't. Lots of other people will think it's a great idea.
Two reasons: First, unless you will be riding at the track, I think a road bike is more comfortable and more practical than a track bike. Second, pistas are a dime a dozen. If you want something to "make your own" it's nice to start with something a little more unique. On the other hand, chrome is sexy.
If you want a new bike, check out the Redline 925 (http://www.redlinebicycles.com/adultbikes/925.html). I think it's a great looking bike, reasonably priced, and not every hipster ******bag™ will be riding one. Stop into the fixed/single speed forum for lots of ideas, opinions, and options, most of which will be different than this one.