I found the default controls to be really off for use with an Xbox One Controller. I also found that editing the controls in the game was a pain because you couldn't do multiple assignments for the same function.

(I'm trying to attach a pic with the controls layout. Hopefully it works.)

To apply these controls,
open
C:\GOG Games\STAR WARS Racer\data\config\current
find a file called
current_control.map

Open it in a text editor, and replace all the Joystick Mappings with this:

####### JOYSTICK MAPPINGS

JOYSTICK BUTTON=7 FUNCTION=CAMERA_CYCLE
JOYSTICK BUTTON=HAT_DOWN FUNCTION=LOOK_BACK
JOYSTICK BUTTON=1 FUNCTION=THRUST
JOYSTICK BUTTON=2 FUNCTION=BRAKE
JOYSTICK BUTTON=9 FUNCTION=BOOST
JOYSTICK BUTTON=HAT_RIGHT FUNCTION=ROLL_RIGHT
JOYSTICK AXIS=RX FUNCTION=ROLL_RIGHT AXIS_RANGE=NEGATIVE
JOYSTICK BUTTON=HAT_LEFT FUNCTION=ROLL_LEFT
JOYSTICK AXIS=RX FUNCTION=ROLL_LEFT AXIS_RANGE=POSITIVE
JOYSTICK BUTTON=6 FUNCTION=REPAIR
JOYSTICK BUTTON=4 FUNCTION=REPAIR
JOYSTICK AXIS=RY FUNCTION=REPAIR AXIS_RANGE=POSITIVE
JOYSTICK AXIS=X FUNCTION=TURN
JOYSTICK AXIS=RY FUNCTION=LOOK_BACK AXIS_RANGE=NEGATIVE
JOYSTICK AXIS=Y FUNCTION=PITCH
JOYSTICK AXIS=RZ FUNCTION=ANALOG_THROTTLE
JOYSTICK AXIS=Z FUNCTION=BRAKE
JOYSTICK BUTTON=HAT_UP FUNCTION=SPECIAL
JOYSTICK BUTTON=5 FUNCTION=SLIDE
JOYSTICK BUTTON=3 FUNCTION=SLIDE
JOYSTICK BUTTON=10 FUNCTION=SLIDE

JOYSTICK AXIS=Y FLIP_AXIS
JOYSTICK AXIS=Z FLIP_AXIS
JOYSTICK ENABLED=TRUE

Pic didn’t work.

What do I need to do if I want to play the game with my steering wheel? I'm mainly having a problem with the game responding to my wheel turning. As in: the game doesn't respond to my wheel turning at all.

I tried to configure it in the options, but that doesn't work.

follow instructions by superjoust, then type them exactly the same as this image.
open 'current_control.map' and use notepad++