Knots can be one of the most boring, overwhelming and underappreciated aspects of fishing.
Knowing how to tie a good knot is a piece of equipment that you carry with you everywhere you go. Just like everywhere else in life, there are a couple schools of thought - know all the knots, or know a few knots and apply them to everything. I prefer the former.
Knot strength is measured by the percentage of material strength at which the knot fails. For example, if you're tying the knot in line that is rated to break at 10 pounds of applied pressure, how close to that breaking strength can you get before the knot breaks? Several knots that are commonly used by fishermen only test at 60-70% break strength - crazy right? In the past couple decades we have really come a long way in being able to test knots with increasing accuracy, which has led to a very quiet revolution in innovative knot design. Really nerdy stuff, but we get to cherry-pick the results. Same with fishing line and tippet design, but we will touch on that later.
If I were starting completely from scratch again, these are the knots I would want to have in my arsenal:
Double Surgeon's Loop. - ties your leader connection to fly line - This is the knot I typically use to create loops for connecting my leader to my fly line, assuming your fly line already has a loop formed in it. This is one of those knots that has a relatively low breaking strength compared to the line you tie it with, but for our purposes that's okay because the 'butt' of the leader - the part that loops or attaches to the fly line - is generally much stronger than the tippet end.
Triple Surgeon's Connector - ties a length of tippet to your leader - In either the double or triple format is a great knot to use, extremely simple and decently strong. The downward pointing tag end of the tippet can be used to tie on an additional fly if wanted.
The Palomar knot, - ties tippet to hook - Another of the very strongest knots, and works for basically all applications where you need to tie some version of a hook onto a line. Even works for braided lines, which are literally super slick plastic strands coated in teflon.
The Eugene Bend knot. - ties tippet to hook. Remember how I said that knots have come a long way in the past couple decades? This one snuck up on me - I only just now heard about it in a podcast that I listened to last week. It's a very strong knot, testing at around 100% strength, and seems super easy to tie. Only drawback I see immediately is that it may go through a little more tippet per knot than others. Not a big deal.
The Non-slip loop knot, ties tippet to hook - Useful for allowing larger flies to move more freely, especially in saltwater on heavier line. You won't need this one immediately, but it's a good one to turn to later.
There are dozens of other knots that can be used, along with some specialty knots that are trotted out for very specific situations to make a fly act a certain way in the water (see the Riffle Hitch, if interested) but otherwise these will pretty much always take care of you.
The best way to learn knots is to simply take some fishing line and brute force repeat them over and over until they're burned into your brain. Fun, right?