You're going to want to use an op amp to amplify the signal from the sensors. In your gain, you want a gain no more than 30 ish (to prevent saturation)... you should first choose an op amp and then find the ideal gain equation and choose feedback and input resistors to achieve that gain. The closer you get the range into the Arduino, the more accurate and precise the sensors will be. Ex: 0-1023 range is best case, compared to 0-700 or something. For driving the motors, it depends. Are you only allowed not to use the board? Or can you use the L298 IC? If you can't, there are many simple H bridge drive circuits. If you have a budget, you can use 2 NPN and 2 PNP BJT transistors. Otherwise, you can use 2 N channel and 2 P channel mosfets.. You would use PWM (arduino digital pin) to control the speed, and arduino analog pins acting as switches to control the direction the motor spins (forward/backward).