Buy an old Gixer and grin like this!

 

There's something to be said for a lightwight kit car. Have been round a track in an old sylva striker with bog standard 1.6 crossflow, with a well balanced chassis still gave silly corner speeds and a whole lot of fun for not much cash.

If you want more speed, spend a few k and get something with less weight and more power. My brother has a Fisher Fury about 550kg, with 180bhp Zetec. 0-60 in v. low 4s, grip at caterham level, took seconds a lap out of any Elise we could find. IIRC, cost him something like £8k to build, might get £6k if he sold it now. Good on sunny weekends, and the kit scene is even friendlier than the scooby scene.

Now all I need to do is learn to drive the things properly

Find someone who knows the cars and you can pick up some rapid and fun machinery second hand for sensible money.