Any Arduino will do. A single stepper will require at most 4 logic lines, even done simply your buttons take a I/O each, and limit switches (top and bottom) each take one.. and that's with just a switch per input, which is wasteful IMO, but.. total of ten.
You are going to need a stepper driver circuit, which type depends upon if it's a Unipolar or Bipolar stepper.
For each stepper you want to control, depending on your circuit, you will need either two or four logic I/O lines. The four line method provides for finer stepper movement, IIRC.
Here's the thread where we are discussing this at the moment. My current design uses Unipolar steppers, with a stepper driver circuit built from discrete components. UL2003 is another common driver, a google for "stepper driver circuit" will yield quite a few options, most of which will work as long as they accept logic-level input.http://arduino.cc/forum/index.php/topic,60571.0.html
Get familiar with the stepper library in the playground, most of what you'll need is there!