For the drive motor, it's easy: Ignore the black wire, and send the blue/brown wires to the first motor output of your controller. When you tell the controller to "go" on that channel, the motor should go :-)
If the servo is non-standard, I don't know how to control it. There are a few ways to run a servo (or even a motor) as a steering thing with only two wires that I can think of:
1) Use a spring that makes the wheels turn one way. Use a regular motor to counter the spring, with stronger PWM the more you want to counter it. If this is the case, you can control that motor from one of the outputs of your controller.
2) Use a centering spring, and give the motor some power to turn either way. Again, if this is the case, you can run it with your motor controller.
3) Use the wires for power and control, and send ground through the mounting chassis. As the chassis looks to be plastic, this is unlikely in your case.
4) Use a digital modulated signal on top of the power signal. If this is the case, give up on that servo and mount your own, because trying to work out something that can control that correctly would be way too much work AFAIK.
Because turning is easier than propulsion, the wires are thinner, because the "steering motor" (whatever it is) probably is a lot weaker, so less current is needed.
Try measuring the motor with an oscilloscope or volt meter when it's turning (assuming you can still hook it up to the old controller...)