SHIFTING: THE BASICS
Shifting is pretty straight forward, although there are some simple tips that can assist in prolonging your drivetrain, and increasing your riding skills.
First off, if you have shifters which come equipped with little numbers in a view window, don't bother with them. Over time it will do you much better knowing by "feel" which gear you should be in a what time. Plus, it keeps your eyes on the trail, rather than focused on the tiny minuscule shift numbers.
Shifting should never be done using force. When shifting, try to release some pressure on your pedal stroke, make your shift and continue on. Gripshift users: try to refrain from shifting a load of gears at one time. Although, these shifters are made to handle these types of stresses, it is much easier on your bike if you shift gears one at a time whenever possible. (not necessarily every time though)
Do not extensively cross your gears. This means that you shouldn't run a gear that puts tremendous stress on your chain. For example, having your chain on the largest (front) chainring and on the largest (back) cog. This gear combination stretches your chain to its limit, and should not be used.