Which motor is best suited to your requirements? I'm going to assume that this is a "robot wars" style device. If not, the explanations below may still help.
The correct selection depends entirely upon the nature of applications for those "several motors" you say you'll need. Each of the motors you specify has several subtypes, and each has its particular strengths and weaknesses. Since you say power isn't an issue (although I think we can discount any AC motor, since I suspect you're going to operate "unplugged"!), efficiency isn't going to factor into the answer.
1) DC motor
Could be a simple brush type, could be brushless. The latter, with a proper controller, is a nice way to get a precise constant speed. I doubt that is important to you in any of your several applications, and the drivers will set you back a few bucks, and home brew designs aren't trivial.
A simple brush type motor is fairly inexpensive to control, but speed often will vary with load. Easily reversed, and easy to set up for dynamic breaking.
2) Servo motor
A nice way to get an especially accurate position control. Probably not important in your application. When purchased separately, the sensors that allow closed loop operation aren't typically cheap, either.
3) Stepper motor
A curious bugger in that while they can provide some good position control (a single optical sensor or switch at the extremes of travel can often be used to set your "home position" references), they can also be run open loop at speeds where the actual steps aren't even noticed and they look like any other continuous motion motor. Just vary the stepping frequency to vary the speed (*really* easy with your processor control). I've found them useful in inobvious ways, and are often worth a look. Driver chips are readily available. Most effective if you can ramp them up to speed with your processor. Under load, they often get cranky about being hit with full speed pulse trains right off the bat.
4) Other Stuff
If you need device that can pound, flip and do other linear or circular moves, don't discount the use of linear or rotary solenoids instead of motors. These are the easier to drive than a motor, and are especially useful for simple linear moves. You'll need to pay specific attention to throw distances and forces generated to be sure you've selected correctly.
Edit:
After seeing your requirement more clearly, the stepper will be of real help to you. You can control both speed and position with reasonable accuracy with these without a lot of expensive rotary encoder hardware that would be needed for closed loop operation. Providing you select your stepper for the appropriate torque (so that it doesn't miss steps due to the load), and use one or two sensors to assure your starting and maximum positions, you should be able to run these open loop and have a very good idea of your position at all times.
Another edit:
Just curious about your suction cups...
You'll need to be creating and releasing vacuum with those, won't you? Do you already have a pump chosen to accomplish that? How had you planned to switch vacuum to the individual cups, and how had you planned to release the vacuum on each? I know that's not part of your original question, but if you can "add", I'm curious.