Bicycle newsgroups are awash with little programs to compute gear ranges, ``gear inches'', and stepping recommendations. I'll take the practical approach.
First, the ratio of front steps to back steps is important. The standard chainring set is 42-52 (teeth). I think of this as 24% step (1-(52-42)/42); if you shift on the big chainring the bicycle goes 24% faster at the same cadence (pedaling speed). The differences for the rear cogs is much smaller. For example, my usual cassette has cogs 13-14-15-16-17-19-21-23. This works out to 8-12% steps, so together with a 39-53 chainwheel set (36%) a front shift is equivalent 3 or more rear shifts. This is unusually large.
If the front-to-rear shift ratio is 1, you have too few distinct gears because for most front shifts there is an equivalent rear shift. A ratio of 0.5 is called ``half-stepping'' because front shifts insert another gear between most rear gears, at the expense of frequent double shifts. A ratio of 1.5 is called ``alpine''; it offers most of the fine gearing of half-stepping but extends the range at the ends, at the expense of even more shifting. My rather extreme ratio of 3 has a different goal; I rarely shift in front and use the small chainring for normal riding and climbing and the big chainring for high-speed downhill or flat-terrain riding.
Modern chainwheels and cogs cannot be combined arbitrarily, they are designed to work together to help the chain move from one cog to the next with small indents and protrusions that ``take over'' the chain and precisely chosen places. I usually pay little notice to this and put my cogs together any way I feel I need for the tour I have in mind, the loss of smoothness is negligible. Effectively, one chooses a rear cog cassette by choosing the extremes, such as 13-23 or 14-32. This decision depends on the type of riding:
* Choose a small range such as 13-23 for flat-terrain riding, and for group riding. When riding in a group, you can't control the speed unless you are in front, so finely stepped gears allow maintaining the optimal cadence for any speed.
* Choose a large range such as 14-30 for mountain climbing. Since the smallest chainring in a two-chainring set is usually 39, you need a large inner cog. I do not like three-chainring sets because they complicate shifting and spinning up hills at absurdly low speeds and high cadences are actually more work than to pump a higher gear.
* When I do a tour than involves both kinds of terrain, I often put together a cassette with small steps for the outer five cogs, like 13-19, and then continue with three extremely widely spaced cogs, like 23-26-30. This looks weird and the big gaps don't shift as smoothly, but I am at home in all terrains. Modern shifters are quite forgiving.
Cadence, or pedaling speed, is an important factor here. It is measured in revolutions per minute. Suppose you ride a road bike with a wheel circumference W of 2130 mm at cadence C, using a front chainring with F=42 teeth and a rear cog with R=17 teeth, your speed at C=100 is
(C * 60) * (F / R) * (W / 1,000,000) = 31.6 km/h
For mph, divide by 1.6. Beginners tend to ride at cadences of 60 or 70 rpm but you should aim at a cadence of around 90 or 100 rpm because it takes less strength and is easier on the knees. High cadences require clipless pedals (a term that means click-in pedals such as Time, Look, or Shimano SPD) because they allow you to apply constant force at all times, rather than stomping the pedal on the downward half of the circle. A bicycle computer that measures cadence is useful to train yourself to use a given cadence, but once you got used to it you won't need the cadence computer any longer.
See also an article on "Pedals"
