To convert a DC motor into something that will work with the project, you would have to get gear the output shaft down a lot.
DC Motors have very little torque and very high speeds, so you need to use gears to produce the required torque.
Also, to have bidirectional control of a DC motor, you need to use H-bridges. These allow a microcontroller to reverse the voltage across the motor in order to change the direction of spin.
Here the SoR tutorial on H-Bridges -
http://www.societyofrobots.com/schematics_h-bridgedes.shtmlIf you made a geared down DC motor with H-Brdiges, however, you wouldn't be able to just slot it into the circuit and it would work.
You would need to learn how to output Pulse Width Modulated signals (PWM signals - also on the H-bridge tutorial) from your microcontroller.
This would be quite hard for a beginner. Anyway, the gearing and H-bridge components would not be significantly cheaper than buying a standard HS311 servo.
I recommend just getting a couple of servos. They are greatly reusable and good value for money.