For fly fishing:
Clinch knot, attaching your fly to the tippet
Double surgeon’s knot, attaching tippet to leader
Nail knot, attaching backing to fly line.
NB there are many possible variations. I have successfully used this one and this one but you may need to experiment.
Another possibility is the Farrimond Friction Hitch.
Two most important knots:
Reef knot (so that you know the difference between a reef not and a granny knot)
Clove hitch (for knotting a string/rope to something) (Once you’ve got this, you’ve got the basis of the tautline hitch)
Third knot depends on what you do: fishing will want a fishing knot. I did rock climbing, so it was the “figure eight” knot. For a lot of hands-on guys, as mentioned above it’s the truckie’s hitch If you’re a surgeon, it used to be the (one handed) “surgeons knot”
Since I’m a figure-eight guy, I don’t use the Bowline. My other personal favorite is the Marline Spike Hitch. When you understand the simple knots like the Reef Knot, The Clove Hitch, and the Marlin Spike Hitch, most of the others (taut line, truckies, Bow, Clinch, and so on) just becomes variations
I learned both in boot camp and then again in electrician school. When you’re hoisting a cross-arm to the top of a poll, the guy on the ground throws a half-hitch around one end of the arm, and ties a clove hitch on the other end. This keeps the arm level while hoisting it up, and allows the guy at the top to maneuver it into position and then easily disengage both knots without struggling. The bowline was used for pole-top rescue, to lower an unconscious person to the ground; if you get it wrong, you cut off the person’s breathing.
You really should try this app. It plays a “video” of the knot being made, which you can slow down or speed up, and allows you to rotate the knot 360° to view at various angles. Should be much easier for you to visualize than drawings in a book.
I’m not counting square knots or half-hitches, but you definitely need to learn those simple concepts.
After that, my Top 3 are:
Bowline
Prusik
Truckers Hitch
Ninja’d. Happy fishing my friend!
Last night I had a dream wherein I was trying unsuccessfully to tie this knot. 
While I’m an Eagle Scout and all that stuff, I can’t say that I’ve ever tied very many knots in real life outside of fishing. When I have, it’s almost always been double half-hitches and the like- attaching a line to something. Generally if I have to attach lines to each other, it’s the good old square knot.
Fishing-wise, I’m a fan of the Palomar knot where I can use it- it’s easy to tie and extremely strong as well. If you do it right, you can tie a leader on with it, and leave a long tag-end with a sinker attached to it, for a simple bottom-fishing rig. If I can’t use the Palomar, I usually default to the improved clinch knot to attach terminal tackle. Occasionally I’ll dig up the Rapala knot if I want a terminal loop also.
I’ve made good use of knots for sailing and for setting up or jury-rigging stuff. Hell, my lawn tractor got through last year thanks to some careful knot tying of the mower deck into the right place. In setting up 10’x20’ canopies and tarps for festivals, a large variety of knots came into play. Especially a monkey fist and then lots of bowlines.
If you do actually start reading the book and learning about the knots, you will naturally start finding uses for them. In addition to many of the excellent knots already mentioned, looking around here I see the buntline hitch, Turk’s head, overhand bend, harness bend, girth hitch, espalier knot, diamond knot, wall knot, Matthew Walker, bottle sling, wall knot, pile hitch, anchor bend, marlinespike hitch, lashing, fisherman’s bend, figure-8, double and triple overhand knots; and, no, I don’t live on a yacht.
Natural selection ensures you will end up practicing and memorizing the most useful (to you) knots by dint of using them frequently.
As an Eagle Scout and Sea Scout in the 60’s, I used to know a couple of dozen knots.
Since then, these four have proven to be essential for everyday life (including a LOT of camping): Square Knot, Bowline, Clove Hitch, Half Hitch.
I sell xmas trees with the cub scouts every winter, my go to knots are the figure 8 loop and the half hitch.
I tie the figure 8 on one end, wrap it around the tree and roof rack, then bring the twine through the loop and pull tight. Secure with a half hitch. Easy to tie when your hands are cold, lets you tighten it and stays tight.
That is exactly a type of espalier knot as described in the book. Goes to show that one is not trying to memorize arbitrary obscure knots and get to dozens of knots that way, rather become familiar with a few basic concepts.
Midshipman’s hitch, Alpine, bowline, figure 8 retrace, clove hitch
I use the alpine as the top of the truckers hitch. The midshipman’s great for tightening when slack, but not slipping under load.
Four in hand
Half Windsor
Windsor
Because one matches the width of the tie and the knot to the width of one’s lapels
That’s how we hoist pipe up to the grid: a clove hitch around one end and then use the length of rope to tie a half-hitch on the other end. This lets the pipe ascend vertically in line with the rope.
Are you and Chefguy talking about an arrangement like #2151 or #2162 [Ashley’s numbering] where the object is hoisted perpendicular to the ground, or something along the lines of #2170, #2181 where the object remains parallel to the ground? Because what Chefguy described sounds a lot like 2151, but he says the spar is supposed to remain level. I have hung up a spar (for a cat gym) the way he suggests, except with two clove hitches, for easy adjustability, before lashing it in place.
They also have an app, which I highly recommend.