You get pink slip cars after specific events.
Palm Harbor Rally - tier 4 pink slip car, Lotus Elise is fastest
Versus - tier 3 pink slip car - Audi TT is probably the best of the tier 3 pink slip cars
Tri City Tournament - tier 3 pink slip car - Challenger or RX7 (Audi TT is better)
4 GMAC car jobs - tier 2 pink slip car - Gallardo or GT2
Meet Carmen - tier 2 - BMW M6 (not as fast as other tier 2 cars, too late to be useful).
For tier 4, the pink slip Lotus Elise is the fastest. For tier 3, the Nissan GTR is fastest, but buying it means you'll have to wait longer to buy a tier 1 car, as you won't have enough money. For a tier 3 pink slip car, I recommend the Audi TT, since it has the best handling. For tier 2, the pink slip Gallardo or GT2 are fastest. For tier 1 there are no pink slip cars. Cars going airborne over bumps and jumps is an issue for the faster cars, and since the Zonda has the most downforce of the tier 1 cars, it's the fastest. (The CCX is a bit faster, but it's not a career car.) Also driver skills increase power more than they increase handling.
As for performance parts, you could use the Elise with race parts in the tier 3 events, but that would mean using just one car for about half of the career mode, so I recommend Elise with street parts and Audi TT with race parts (or the GTR if you're willing to wait longer to buy a tier 1 car). Pro parts get unlocked about 6 events before you get a pink slip tier 2 car, so it's not worth it to put pro parts on a tier 3 car.
Note that "current" races on the GPS map don't include races more than a few levels old, and unless you switch the mode to "available" you'll end up skipping some races because the GPS map won't show them in "current" mode.