How to avoid wearing out gears? I have a Specialized Tricross which I bought new last year. 

In that time (approximately 1K miles), I wore out my chain and gears. At least two of my gears are still slipping, even with a new chain.

I think I spend the majority of my time in one or two gears. Should I avoid this?  Are there other things I should be doing to avoid replacing my chain/gears every season?

I am a novice who mostly just commutes to work, so assume I know very little about form.
A stretched chain will wear out gears (especially the rear cassette). You can see when this because the teeth get worn away to points - it's very obvious.  

But it's very unlikely you wore out a chain and gears in 1000 Km. What is more likely is that the cable has stretched slight from new and the derailuer is out of adjustment so it is putting the chain slightly between two gears.   For adjustment details see - http://www.sheldonbrown.com/derailer-adjustment.html

To answer your question - No you aren't going to do any damage by using only one gear, if it's the correct gear. You are going to stretch the chain, and break more of them by being in too high a gear - just like in a car. You will also wear the chain faster by being in a 'bad' gear - eg on the inner front cog and the smallest back cog or (v.v.)