There are 2 stages to upgrading your car: R&D Center and Garage. You need to perform tasks in BOTH of these buildings before your car upgrades are complete.
Step 1: Head to your R&D Center in your Fan Zone (see Fan Zone for more info). Once inside, you will have the option of researching “Upgrades” for your 5 car parts. Select the part you’d like to Upgrade and purchase it using in-game cash or gold.
Step 2: Once you have researched an upgrade, you’ll need to install it on your car or cars. Head over to your Garage. Once inside, select the car you’d like to install the newly researched part on, then select the “Installation” tab. Tap the part you wish to install and start the timer.