Although I personally know many more knots and use them on a regular basis, when I'm teaching for my school's Outdoor Education department, I only teach 4 knots for backpacking:
Bowline (One end of a tarp ridgeline; attaching ropes to our bear bags, which have a loop for this purpose)
Trucker's Hitch (Tensioning tarp ridgeline and stakes)
Sheet Bend (Tying two lines together, primarily if the bear bag ropes are too short)
Tensionless Hitch (Tying off the bear bag ropes, since we don't use the PCT method)
And one of those isn't even really a knot... the tensionless hitch is literally the easiest thing I've ever taught.
Basically, as others have said, figure out what you need to do that includes knots, then find the knots that make the most sense to you for those purposes.