The best upgrade parts are:
Traction - R-600 Repulsorgrip
Turning - Control Stabilizer
Accelerate - MAG-6 Injector
Top Speed - Block6 Thrust Coil
Air Brake - Quadrijet Air Brake
Cooling - Turbo Coolant Pump
Repair - Cluster2 Power Plug
A couple quick tips to save up enough money, cause some of the parts are pricey:
1) Set the stakes at winner take all. You're probably going to win the races somewhat easily, especially the early ones, so you may as well get paid extra for it.
2) Invest in all 3 pit droids ASAP. This will help ensure you don't have to replace broken parts very often, thus saving money in the long term.
3) Use and abuse the Junkyard. You get often better or comparable parts to what you have for less money, and they'll easily be repaired, especially in the amateur and semi-pro circuits. And sometimes you can even get money back while getting a "worse" or more damaged part.
Personally, I found that the best parts are more likely to appear in the Junkyard, so always be sure to look in both places!
Edit:
If you've already beaten all the races, you can make more money by buying worse/degraded parts that what you have, though that may be a long process.
It may be faster to start a second racer profile, and just burn through early races quickly to save up. However, upgrades don't transfer from profile to profile.