First of all, as a steampunk who is just getting into Arduino programming, welcome.
Now from your message, it sounds like you are new to programming, etc. If you are not, feel free to ignore all of this blather.
Before going on, I would suggest you think about all of the effects that you might want to put in the project.
- Do you just want gears moving all of the time during the show, or do you want a gear moving and then stopping based on some signal?
- Do you want gears to speed up/slow down/reverse direction, or do you just want the gears spinning all of the time?
- Do you want several gears going at the same time?
- If you do have several gears going, do you want them programmed independently, or will all gears operate together?
- What is your budget?
- What is your time frame for building the project?
- How long do the gears need to operate, which in turn determines how much battery power you need?
- How much mechanical skill do you have? How about programming skill?
Generally with projects like this, you don't want a big bang theory where you need to have all effects working all at once, but you want to start smaller, and add effects one at a time. This way, if you run out of money or time, you still have some working effects.
In case you don't know what they are, a servo is a motor is restrained to operate between 0 degrees and 180 degrees. You program it via the Servo library, and tell it to move the horn to a particular degree. For example the Knob example sketch takes a potentiometer to control where to move the arm: http://arduino.cc/en/Tutorial/Knob
Here is a video on how to use servos to control radio control airplanes wings that you might find useful:
Now, I was curious and did a google search on 'lilypad arduino motor' and I didn't see much out there, other than this example using a servo to control a pillow:
Now, there are servo motors that have been modified so that they operate continuously. These are programmed the same way as Servos, except if you give it degrees 0..89 it rotates in one direction, and if you give it degrees 91..179 it rotates in the other direction (and the 'degrees' passed control the speed). I imagine these are what you want to use, and they only need one wire connected to the Arduino/lilypad for programming and ground/power. Beyond the simple continuous servo motors, there are stepper motors that give you more control, but the programming is more complex. Here is a simple video on continuous servo motors:
In the Arduino, I have both a LCD shield and a servo, and I find that I can't use these together, because the Servo library disables PWM on certain pins that the LCD shield happens to use. I don't know the details of the lilypad to say whether there are similar restrictions.
One other thing, servos and motors tend to be rather power hungry, and often times you need to consider wiring the motor with separate power supplies from the Arduino/lilypad (you do need to connect all of the grounds together). Here is a site that says how to wire up servos: http://rcarduino.blogspot.com/2012/04/servo-problems-with-arduino-part-1.html