Moving a large body is theoretically possible, of course, but there is only thing required by Isaac Newton: You need to eject mass away from the planet.
Simply sticking a rocket nose-down in your back yard won't work, for two reasons:
1. The rocket's exhaust will loose all it's momentum after traveling up through the atmosphere for a few hundred feet.
2. Even without an atmosphere, a chemical rocket's exhaust cannot reach Earth escape velocity, and thus all those gas molecules roaring out the rocket's nozzle will fly up into the sky and then fall back to the surface. The exhaust mass must escape the body's sphere of influence in order to create thrust.
On the moon, of course, point number 2 is the only one you need to worry about.
Now, if it makes you feel any better, we have moved planets, including the Earth and the Moon, already! Every time we launch a spacecraft and inject it into an escape trajectory, we have expelled part of the Earth's mass in one direction, creating an equal and opposite reaction in the other. (Which, of course, is so small that you can't even measure it, so it's really just a thought experiment.)
We've also launched and landed and crashed objects on the moon as well, and we've performed slingshots on all the other planets save Pluto, which will get a flyby visit in a few years.