Can I use the drive train of another bike? I was gifted an old bike that I've decided I want to fix up and use. I believe it's a hybrid, not quite road and not quite mountain bike. It's 21 speed and it's an Apollo quick cross (I can't find anything about it on the web, have emailed the manufacturer asking for the manual).

Despite knowing nothing about bike maintenance I can tell the drive train is in fairly desperate need of replacing (missing and worn teeth on the chain set). The idea I have is finding a donor bike and borrowing its drive train. Is this possible and what should I be looking for in a suitable donor bike?
It looks like a relatively basic hybrid based on a quick google search. 

You can probably find another bike and get the parts off it and onto this bike, but this is almost never cost effective (you either need to borrow/buy the special tools you need for removing cranks, replacing the chain, etc. and the time to learn how to do this or pay a shop to do it, and the labor at a bike shop won't make it cost effective). 

I'd suggest having someone look over the bike to make sure its safe to ride (is the headset loose, are the brakes doing their job, is the frame cracked, etc.), since it sounds like this bike has been neglected. A bike shop should do this for a nominal fee and tell you how much you'd need to get it into decent condition. You may find that for the cost of repairs (labor + parts or time investment) may not be worth it for you (and you may want to sell this bike and get something else). 

Without knowledge of the bike, there are several things that are commonly needed to recondition bikes:


Cables + housing for shifters+brakes
Brake pads (sometimes the whole brake is more cost effective if the old one is in poor shape)
Shifters (they wear out/break)
Freewheel/Cassette (gears in the back)
Chain
Truing the wheels
Repacking the hubs of the wheels


And then various adjustments (e.g. headset). Crankset+BB in some cases as well. Note that the crankset might not actually be damaged -- some teeth are designed to look worn out, but without a picture of the bike or something, we can't even begin to make some sort of assessment. 

Most of these require bike-specific special tools, so if you're not inclined to take up the maintenance on your own regularly, you're best off finding a shop to look it over.