Most rear cogs have their number of teeth stamped on them, or you can just count them.

As you are riding, just click up to an easier rear cog if you are pushing hard, or down to a faster cog if you are spinning the cranks too fast. You don't need to know pedal rpms or which combination of gears to use at each speed.

Anyway, here's more info to help visualize how it works:

I like to use Mike Sherman's gear calculator. It created this chart. It'll change the numbers on the fly as different cogs and chainring sizes are selected.

The chart shows the front's big chainring in black (50 teeth), and the small in red (34 teeth). Each bar is the speed range you get from pedaling between 80 rpm and 100 rpm. Usually the most efficient pedaling is in this range. I try to stay around 90-95 rpm, the right side of these bars.

For example: See the 15 mph vertical line. You could be using a 50-21 combination, pedaling about 80 rpm, or a 50-24 at about 90 rpm. Or use the small chainring with the 34-15 or 34-16. Which combination mostly depends on the road ahead. If the road is flat or downhill, the big chainring will let you go faster. If it'll be uphill, switch to the small chainring.

If it's really steep, you'll be in the easiest combination, 34-27, and pedaling at a much slower rpm, perhaps going only 4 mph. ( This is called "mashing" as compared to "spinning" above 80-90 rpm)