Catrin:

Two quick notes: The derailleurs only shift when the drive train is in motion (e.g., while you're pedaling); and they won't shift well if the drivetrain is under a heavy load. So, when you hit the shifters, keep spinning the pedals for a time or two until you feel it shift, and if you're cranking up a hill, ease off the pressure on the pedals until the gears shift, then you can pick back up again.

The big idea about shifting gears is to be able to keep up about the same pedal pressure or level of effort, as you roll along, even going up or down hill. You shouldn't have to be mashing a great big gear unless you're into strength training; drop down a gear or two until you can spin alongwith about the same pedal pressure as before you hit the hill, only at a higher cadence. Let the mechanical adantage of the lower gear do the work for you...