Your right that you will/could learn a lot designing and building your own servo. One can of course just need a functional model servo for a project and get them really cheap these days, $10 and up, maybe less on E-bay. As far as the motor, a simple DC motor will work, rated to the voltage you have available to drive it. You will need bi-directional control so some kind of H-drive circuit is called for. There is a need for position feedback to the controlling electronics. This is usually pot mechanically attached to the gear train to measure actual position of the servo output. The controlling electronics takes the feedback measurement and the desired position command and decides if there is an error between those two measurements and decides what direction and how hard to drive the H-drive motor driver circuit. A common control method is to utilize A PID algorithm.
This can all be quite complex for a first time project or for a beginner. And of course you will be spending tons more for the parts and will never make it as small and compact as a store bought servo. However the reward of learning how a servo actual works and getting your own design to function can be worth a lot.
Lefty