Did you at least try to run your 12V motor from the servo?
Try connecting the string that should lift the cabine to the second gear or maybe the third one
And figure out each story is reached at which angle of what used to be the 'output shaft'
then for each level you'll continuously send a different signal to hold the cabine at a certain level or to go to a certain level (thus the need for switches or look up some fancy toggle circuitry)
You can't just keep thinking about this and we can't give you everything(it's not possible to predict what you'll run into), you just have to start buying parts and figuring out how things will work together
If you're really stuck, start making the chassis.