-You must pedal in order to change gears.  When changing gear, pedal 
 lightly.  It will save your drivetrain from wear and tear.
-If you have "numbers" on your shifter, don't use them.  Instead, 
 calculate the gear inches and use that as your shifting guide.  You 
 should be able to locate a program for this from the rec.bicycle.* FAQ.
-Shift before you think you have to, e.g. climbing.  When you have to 
 shift, it might be too late.
-Do not cross your gears, it will kill it.  This means that you do not 
 run a big chain ring with the large cog or the small chain ring with the 
 small cog.
-Shift lightly on the levers.  There is no reason why you need to press 
 the shifters real hard to shift.
-To save the drive train from wear and tear, make sure it is clean and 
 well lubed.

Some added:
Graham Barnes [barnesg@tierfon.hao.ucar.edu]
I agree about not using the "numbers", but I'm not so sure that calculating 
gear inches is worthwhile.  I've been mtbing for ]5 years, and I've never 
bothered to work out the gear inches for any of the bikes I've ridden, except 
when I was thinking about changing cassettes.  Maybe I'm missing 
something, but I always went with the philosophy that, if it's hard to 
pedal, shift down, and if I'm spinning madly, shift up. 

John Stevenson [johnstev@world.net]
It's very unlikely that anything but water and crud are major factors in
wear of MTB drivetrains.  Sure, in theory careful shifting and avoiding
extreme gear ratios will prolong drivetrain life, but in practice I suspect
that the damage prevented by these practices is insignificant.

[in regard to gear numbers]
I go along with Graham here.  A better reason for avoiding use of those
silly shifter windows is that if you're looking at them, you're not looking
at the trail, which is where your eyes *should* be.

[in regard to shifting before you have to]
I think this can be better expressed:
Shift before you have to.  For example, when you're climbing, shift into a
very low gear as you approach and start the climb.  If you wait until you
are about to stall it may be too late to shift.