Each input method has different pros and cons, but controller is by far the best option. You can enable controller by selecting "Use/Enable Joystick" in the config.exe in the game's install directory, or through Rebuilder if you are using that (Rebuilder overrides config.exe). Any xinput controller such as an Xbox controller should work, Playstation controller users might have to use DS4Windows.

Controller pros: Fastest acceleration, precise turning, can use d-pad to rotate your camera angle without rotating the player (useful for collecting pieces during the piece hunt/Brickster chase). Hitting a wall/obstacle during any vehicle section results in a much smaller time loss than keyboard, your speed is regained almost instantly after.

Controller cons: You'll be using controller for movement, mouse for clicking/pointing, keyboard for spacebar. I have spacebar rebound to a mouse button to make life easier, but this isn't required. Holding straight forward during the car race can be difficult, but you can hold up arrow on the keyboard to lock in your angle to avoid this.

Mouse pros: Similar but slightly delayed acceleration compared to controller, more precise turning than keyboard.

Mouse cons: No free cursor while moving, will be held upwards to move forward which can make collecting pieces more difficult. No camera rotation independent of the player direction.

Keyboard pros: Same acceleration as controller if you hold control while moving.

Keyboard cons: Really awkward and imprecise turning, no camera rotation independent of the player direction.

This video shows the difference in acceleration between keyboard with no control held(left) and controller(right):