I'd go with an S2000. Lower mileage AP2 for $18,000 tops. Add a rollbar, hardtop, coilovers, wheels, tires, brakes. Should still be under $25,000. Eventually get into that to make it faster as I'd gain experience. I'd probably have enough left to build a second car, even with all that done. MR2 turbo, maybe.
Alternatively, an LSx swapped FD RX7 can be had for around $20k. Sexy looks, sexy sounds.
Edit: The issue with the 370Z were that the Nismo package did not include the Nismo track pads. They overheated and basically lost all braking power. Afterwards I believe all Nismo Zs came with the pads as standard or something like that.
Last year an almost brand new 370Z popped a motor at an autocross in Nashville, TN.