Will it require more, less, or the same amount of energy to lift a spaceship off the Earth than the Moon?

Well it will definitely require much less energy, so yes you are right. The atmosphere part is certainly true. The main total reason is that the escape velocity is much lower on the moon than on earth. That is due to gravitational pull (determined by the m of the object), yes the atmosphere does come into it too although to a lesser extent. The real reason NASA would want to launch missions from the moon to other planets is that it would cost MUCH less. If we can effectively harvest hydrogen on the moon, we could create fuel to launch missions for far less :_)