These are hard questions to answer, not because the information you need is hard to come by, but because answering them in a straightforward way will keep you from learning how to do it yourself, which is presumably why you're going to college in the first place. On the other hand, it reasonable to say, "Look, I don't know how to do this, and there is no point in trying to reinvent motor control from first principles."
So how about this for an answer that will get you started.
You're trying to do a task where the motor will need to spin a little bit, then stop and wait, then repeat. Also, overall, this is a pretty low power task-- you're not trying to move a car or train. For this kind of low-power, intermittent duty task, I'd probably start with a small stepper motor, like you find in old disk drives or printers. You could also use a small DC motor, like you find in most motorized toys, but you'll probably need some clever ratcheting mechanism to make the intermittent motion you're after.
In general, motor power is proportional to volume; you want a motor that is bigger than a grape and smaller than an apple.
The next step is to google "stepper motor" and "dc motor controller." I'm afraid Wikipedia will be incomprehensible at this stage; I'd look for tutorials from robotics clubs or intro electronics sites. You might start with:
After that, since you say this is a low-budget project, I'd try to find something with a motor in it that you can reuse-- maybe the mechanism from an inkjet printer (stepper motor) or an RC car (DC motor). You might be able to reuse something without taking it apart-- like maybe you could just have the printer print a page once a minute, and the motion would pull on a string that somehow moved the ice.
From there, draw a lot of pictures of what you want to build and try building some crappy prototypes out of cardboard, string, and paper. You'll learn faster by testing and failing than by trying to plan it all out in your head first.