Shifting the derailleur does change the gear; it's not a multi-step process from the riders point of view. There is a slight delay while the derailleur mechanism moves the chain from one gear to the next, but if everything is adjusted properly, then the delay is fractions of a second.

You do need to be pedalling for the gear to change; the derailleurs rely on the movement of the chain to actually work. If you try to shift while coasting or stopped, then the derailleur will move, but the chain won't; therefore, you don't change gears. When you get going again, you'll hear some pretty awful noises being made, and the chain might fall off.

That said though, try not to shift while pedalling hard. i.e., if you're really pushing the pedals and you try to shift, then you'll hear some awful noises and the chain might pop off. Try to "let up" a bit on the pedalling pressure while still pedalling, and it should shift quickly, quitely and smoothly.

If you continue to experience chain-falling-off problems, then take your bike to your local shop. They can easily adjust the derailleurs; the chain shouldn't ever fall off, but somehow they always find a way to come off occassionally anyway.