From someone who brings in a lot of cars for race/rally use you would be hard pressed to get yourself something better than a Version 5 or 6 STi Type RA. Within limits*, this vehicle will provide you with more fun and time in car for your dollar than you can poke a stick at. JWS recently sold a V6 RA which competed at several hill climbs, with Michael Lloyd driving, absolutely bog standard, running normal road rubber and this car was setting times leaving much more fancy machinery for dead. This car sold for $13k. With a budget of $20k you'd have all your consumables (r-comp tyres, brake pads, fuel) and still have change left over, after a year of club-level motorsport, for a replacement gearbox if you simply cant drive without any mechanical sympathy.

My 2c worth

* Limits being keeping the power at stockish power levels and driving with some mechanical sympathy. Winning at any level of motorsport will cost $$$$$'s.