Knots are essential in fly fishing rigging. In a typical fly fishing rig, knots are used to attach an artificial fly to the tippet, the tippet to the leader, the leader to the fly line, the fly line to the backing, and the backing to the reel. For each of these connections, there are many different knots one could use. While many fishers take great pride in mastering the tying of all the knots used in their fly fishing rigs, fly fishers do not need to know them all.
After 40+ years of fly fishing, I can probably count on one hand how many times I've had to connect backing to the reel or a fly line to backing... and never have I done so on the river. I've always tied these knots in comfort with access to a cheat sheet to refresh my knowledge of the seldom-tied knots. But more often I simply rely on professionals at the nearest local fly shop to tie the specialized knots used for these connections.
Fly fishers need to master just two knots: a terminal knot to attach an artificial fly to tippet and a tippet knot to attach tippet to a leader. For the former, the improved clinch knot will do. For the latter, a surgeon's knot will do. And, as we often use loop-to-loop connections, be sure you master this connection as well. There's definitely a right way (symmetric connection) and a wrong way (one side folded over) to put the loops together.
Only after mastering a terminal knot and a tippet knot should you consider learning a new knot (for these or other purposes).
For attaching flies to tippet, I typically use the uni knot. This knot is also known as the Duncan loop knot or grinner knot. I find it dependable and easy to tie, even in the dark, with minimal material waste. You can even leave the loop open to allow your fly to move more freely and add stealth. Occasionally I'll use a double Davy for attaching fine tippet to small flies for a less bulky knot. For large streamers, I'll use a non-slip loop knot, also known as the Lefty Kreh loop knot.
For attaching tippet to leader, I rely on the double uni knot. I had previously used the easy-to-tie triple surgeon's knot. However, when i started focusing on dry fly fishing, I wanted a knot that would allow the line lay straight on the water. As I had already to the uni knot as my primary terminal knot, the double uni knot was the most natural choice for me. I've found the double uni knot to be quite strong and dependable, even when connecting significantly different sized lines.
Not that I use them all that much. Here's a few more knots that I have in my quiver.
For making loops for loop-to-loop connections, it is useful to know how to tie a perfection loop knot. But if you haven't mastered it, the non-slip loop knot is a good alternative. The bulkier surgeon's loop knot will also do.
I'll use riffle hitches when skating dry flies.
The nail knot is a useful knot for attaching leader material to a fly line when the fly line doesn't have a loop connector.