It won't go to the moon because it does not have sufficient velocity to escape the gravity well. Not because there is no lift in a vacuum (it will never reach a vacuum).

It will simply float to the top of the atmosphere (if it stays in one piece) to the point where the weight of the baloon is balanced by the buoyancy. Just like a rubber ball will not float 'on top' of the water that supports it but slightly inside it.

It's Archimedes' law applied to balloons.