When your upgraded car still is outclassed by a stronger car with better baseline stats, consider purchasing a new, better car and continuing your upgrades on it.
Prestige
Changing your car's color, adding vinyl, and replacing its rims are cosmetic effects that will not change your car's performance. However those modifications will increase your prestige. The higher your prestige, the more bonus money you will be awarded at the end of a race.
Shifting
Each car has different shift timing. When the yellow icon on the tachometer appears, the perfect time to shift in the green will soon follow. If you shift too early or late into the next gear, shift back down and build your speed back up.
Upgrades
Regularly check if your car has any upgrades available. Purchase them if you can afford to do so. Start by getting the nitro boost. It can only be used once per race, but is very useful.
Various Tips and Tricks
Time your shifts carefully. One of the best ways to do it is to watch for yellow on your tachometer, which indicates that you're about to hit the perfect green range.
If you shift too early, shift down again until you reach higher speeds before shifting up once more.
Deck out your car with vinyl and new rims it boost your prestige and that means more money at the end of each race.
Pay to upgrade your car whenever you can, so you don't have to rely entirely on your shifting skills to win races.