"But still: which direction is which? Do I shift to a smaller number going up a hill, or down? How do I know when I should shift it? Where does the normal "D" fit in?"
Many others have covered what to do when, but I always found the explanation of the mechanics to give me the most information. In a vehicle with an automatic transmission, D means normal drive, which will use all of the available gears. The 3, 2 and 1 indicate that you're restricting the gearbox to just that top gear, so choosing 3 will allow it to use first, second and third gear but it won't shift into the top gear. Lower gears mean lower ratio, and that means more power to turn the wheels but a lower limit on how fast they'll turn (at a given engine speed). So, shifting down forces the car to go slower in general unless you step on the gas pedal. That means that going down a long hill, you don't have to use your brake because the engine will top out and keep the car from going faster (as long as you don't accelerate). In bad weather, it keeps the car from going too fast and gives a more constant pressure on the drive train which can help with traction.
So to answer your question, when you need more control, you downshift. Normally when using an automatic transmission, you should never need to downshift for power unless you're towing something pretty heavy since the transmission will downshift when you need it and unless the vehicle is heavily loaded or climbing a very steep grade it'll be able to shift smoothly. Given that, you'd want to downshift on long downhills so that you don't have to ride your brakes, and you'd consider downshifting in snowy weather if you find that you're having trouble handling in normal drive. How far down is a matter of feel and the particular vehicle, so you'd have to try it out, starting with the highest number below "D" that you have and working downward. Most cars will be almost inoperably slow in 1 so you'll probably never use that but again, it's a matter of experience and preference.