Tap on your car in the main menu to enter the Garage. To upgrade your machine, you need gold and spare parts. Both can be found in the supply boxes that you get by winning quick fights and being promoted into the next stage or league.
There are 4 types of spare parts:
Chassis is the "body" of your machine. It has several randomly placed slots where you can install other spare parts. Each part has its own type of slot. Weapon slots are hexagon-shaped, while slots for gadgets have square shapes. You can't place a part into the slot meant for another type of parts.
To install or change the part, simply move it into the needed slot.
You can also upgrade the parts that you already have. To do that, you need to fuse the unused parts into them. To improve the part, first tap on it. You will see its full characteristics in the window that will open on the left side of the screen. While keeping this window open, scroll your inventory until you find the unused part that you are ready to spare. Drag it into the window with the part you want to upgrade and fuse both parts. This will boost the characteristics of the upgraded part. The unused part will be destroyed. You can fuse different type of parts with each other. For example, you can upgrade a wheel by fusing a rocket launcher into it.
Beware, though, that each fuse costs you some gold coins. The price depends on the class of both parts: the better each part, the more gold you'll pay for a fuse.
And don't forget about stickers! Make your machine more stylish by placing them on your chassis.