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Do you want to become a sprinter or enduro? Picking a frame based on looks is not the way to go if you want to take it seriously. Important to note that the bike will not really make you improve or go faster, hard work does that. I sprint and use a DF4, its stiff, decent price and perfect for what I do, most money went towards wheels, components and rings/cogs. What is your budget?

Start with a neutral fitting frame. Nothing one way or the other. The Fuji is a good low cost example. A lo-pro is an exampe of what one should not get.
Important bits are the bars, drivetrain, tyres and wheels. I'd start off with used old school tubular wheels-- can be had cheaply since everyone has fooled themselves into thinking they need deeper section carbon--- and go with a nice drivetrain (crank, chainrings, chain and sprockets) and good tyres. A good round drivetrain with good tyres goes pretty far. A lot you can get used. The handlebars I'd buy new and, again, keep to neutral. Can't, I think, go wrong with the Deda Velocita or Pista fitted to a Deda stem. The first act will probably be to collect chainrings and sprockets. They don't have to be Zens and Superstars but I'd avoid cheap sprockets and rings.. For tyres I'd start off with Conti track tyres as they are relatively robust..

Go to Herne Hill and do the induction with a borrowed bike.

http://www.hernehillvelodrome.com/starting/

Check out Dolan maybe?

Don't buy a lo pro. There are reasons that no-one who races at HHV (or any other track/ World champs / olympics etc) doesn't: fit, handling, etc.
Even though track races are shorter, you are still on the bike for long enough to become uncomfortable if it's badly setup.. and if you go to training sessions, you will be on the bike for the best part of 2 hours.

A Dolan pre cursa with decent parts would be my go-to build for anyone starting track and can be built pretty cheaply. I would echo what Edward has said re: keeping things neutral - don't bother with track specific bars etc until actually feel you need them. I hardly ever use 'track drops' despite leaning towards sprinting.
Alternatively there are lots of great OTP options from Fuji, Hoy, Planet X..

You can upgrade the components slowly and if you're really keen get a fancy frame later. You can pretty much guarantee that if you are racing on one and not winning, it's not the bikes fault! The great thing about track is that you don't need to spend loads to get a decent bike.. and kit lasts ages too.

IMO the biggest benefit you get from moving from loaner bike to your own is the bike is set up exactly the same each time you ride is.

Grabbing a bike out the containers is great but you are very unlikely to set it up with exactly the same measurements each time.

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