As a climber, fisherman, and backpacker I know a good deal of knots.
You can find instructions on the internet for most.
I think if you learned the following you'd be covered for just about everything you'll do backpacking:
1. Bowline (strong loop, unties easy)
2. Figure 8 on a bite and figure 8 follow through (one of the strongest loops, probably more than you need for backpacking)
3. Clove hitch (good for tying off to stakes, etc. frees easily, very quick to tie.)
4. Taughtline hitch (sliding loop, allows you to tension a line)
I could go on and on, but when I'm backpacking, these are the knots I find myself using the most- can't think of the last time I used something else (unless I was fishing or climbing)
I love knots!