Sebach

Congrats on the new ride!

Your chain shouldn't be falling off when you're in the middle ring up front and on the smallest cog on the rear. You might get a bit of chain noise but that should be all. It could be other things (component wear, chain stretch, etc) but that doesn't sound likely so I would guess that you probably have a derailleur adjustment problem there. Easy to fix by yourself with some time, a couple tools, a guide of some kind (website/book/friend) and some patience.

If you go to MEC (there's one in Vancouver), you can check out a book called "Zinn and the Art of (Mountain or Road) Bike Maintainence." Two years ago I was exactly where you are now but that book, a bunch of tools and a dozen skinned-knuckles later... I can build a new bike up from parts. I'm not a pro mechanic by any means but when I'm on a long ride, I like feeling more secure just knowing that I can fix something if it goes wrong. That being said, I still swing by my LBS every now and then to get them to do some things I don't want to mess up or don't have the tools to do (cutting headtube, pressing cups, etc).

Collect some tools, gather some resources (books/websites/etc) and dive in. You'll be a lot more secure knowing how to fix your ride and you'l be a lot more confident knowing that you can handle most problems if they should come up. Don't worry too much because if you mess something up or can't get it back together, you could always 'walk' your bike down to the shop to get them to fix it. It might cost you a bit of money and pride if it doesn't go right but the possible benefits and experience far outweighs the downsides.