There are so many fly fishing knots available to learn that a new fly fisher is often overwhelmed by the options and has a bit of trouble deciding which knots to tackle. Experienced anglers often overlook the difference a knot could make in their presentation or numbers of landed fish, not to mention the effect a knot can have on the numbers of flies you keep/lose.
If you are new to fly fishing concentrate on learning a few basic knots. For Trout you need to learn knots that will connect backing to a reel, backing to fly line, fly line to leader, leader to tippet and tippet to fly. That sounds like a lot, but it really isn't too bad if you break it down. Here are my go-to knots for most Trout applications:
Arbor Knot - backing to reel
Nail Knot - backing to fly line
Nail Knot or Needle Nail Knot - fly line to leader (non-looped fly line)
Perfection Loop - leader to fly line (looped fly line)
Blood Knot or Surgeons Knot - leader to tippet
Clinch or Uni - tippet to fly
If you've mastered the basic connections consider how changing knots might improve your fishing. For example, if you fish streamers in a lake or river would an open loop knot such as a Non-Slip Mono Knot or Rapala Knot allow your streamer to have more action. In most cases the answer is yes. Another alteration is needed when chasing large fish on light tippet. In this situation consider substituting your normal, quick to tie Clinch or Improved Clinch for a super strong connection like a Palomar Knot. The Palomar is the strongest knot I know of and is quite simple to tie. The only drawback I can think of to the Palomar is that it uses a bit more material to tie, so you may burn through more tippet if using it regularly. Another great time to evaluate connections is when using long leaders (dry flies or Euro-nymphing) or fishing with little fly line out of the rod tip (small streams). In those situations it is common to frequently bring the fly line to leader connection in and out of the rod guides. A bulky, uneven connection will hang up in the guides and cause you to loose fish as well as drive you crazy by making you stop fishing to pull the fly line back out of the guides. Try a Needle Nail Knot or Superglue Splice to create a smooth connection that easily slides through the fly rod guides. Welded loops are available on most floating fly lines these days and although they are convenient, they're bulky and don't slide through the guides very well. It is painful to pay nearly $100 for a new fly line and then hack off the end of the line but in many situations doing so will make your time on the water more efficient.
Here is a cool website dedicated to knots:
Grab some line the next time you park it on the couch to watch a game or TV show. Spending a little quality time practicing your knots can only improve your fishing. I know it has mine.