Do only the drag races until you complete it. You should now have at least an RX-8. The rest of the game should be very easy.
Easy wins
The key to winning on any course or mode is to have a flawless race. Avoid traffic, cut corners, and drive straight at every opportunity. A tip is to check where you will be racing next and set up a custom free-race to emulate the race course; here you can check the feel of the course and find valuable shortcuts.
Quick boost
If you crash your car in a race and you have a manual transmission, put it in neutral. Hit the gas all the way, then slam it into first gear to get a very quick boost.