To improve your overal times try to race with the braking assist on at least low as it will make a huge difference in your speed when cornering.
Look for corners in the track that you can drive across to flatten out the curve so you can go through it faster than you normally would.
If your vehicle has been damaged you will notice a decline in it's performance, the only way this can be fixed is to put the car in for repairs.
Make upgrading your first car your priority as you will not progress very far with the standard build.
An easy way of earning money so you can get more cars and upgrades is to replay races you have already finished and know your way around.
Ignore the resprays as they are expensive and do not improve the performance of the car. Just leave your cars the default, random colours.
You can sometimes save a little money by not repairing or postponing maintenance. There are plenty of races that don't require that a car be at full strength to win.
Shorten Wait Times
If you'd rather not wait for time to pass before you can upgrade your vehicles, there's a way to get around that restriction. You'll have to adjust your device's time settings, however, and manually adjust the date forward. That will trick the game into believing that the required time has passed, and you'll receive the desired upgrade without waiting. Then you'll want to return the device's system clock to its normal time so that you don't have issues with other applications.
Only upgrade when it is necessary so you conserve your money. If you do have to upgrade, just upgrade one car fully in each circuit.