Purchase the Quarterback for 14000 cash, and max out the upgrades. You'll need about 25k in total
The first Time Trial is the easiest one:
Derby Initiate Series -> Bruxton River Run Trial.
Gold medal is a time under 1'22"
With one of the first 2 cars available to purchase from the game (Grosse Pointe and Derbie Cookie) I was not able to set a lap under 1'24" but with the Quarterback on the first try I did 1'15" without much effort.