I'm going to start off with the very basics, and go to more advanced jump mechanics and rollouts further down. I will only be covering jumps that are useful in actual games, so I won't be teaching you jumps like syncs and wall pogos. To learn the jumps that only apply to jump maps, hit up TF2Jump.com or YouTube. To practice the following jumps, load up a map and enable cheats by typing "sv_cheats 1" in console, followed by "hurtme -99999". If you need to refill ammo type "impulse 101".
I find the best way to learn anything is to take it a step at a time, so I'm going to break down learning the basic jump as follows:
Start by aiming straight down at your feet. Jump up, then shoot the ground under you. Repeat this with different timings between jumping and shooting until you've got the timing that gives you roughly the most height.
Once you're comfortable with jumping and shooting, add crouching to the mix. It goes in front of the original two actions, so now it's crouch-jump-shoot. You should notice a considerable increase in height. Try to keep the delay between crouching and jumping minimal, because if you hold the crouch in for too long, you won't go anywhere.
When you can comfortably do the standard rocket jump straight up, it's time to practice actually going somewhere other than straight up. It's as simple as aiming slightly behind you (all the way behind if you're using the Original instead of the default launcher), and then doing everything in step 2. Once you start turning, switch from holding W to A so you keep going in the same direction. As soon as you're airborne, snap your mouse back to see where you're going.
Now that you're soaring majestically through the air, it would be a good time to incorporate airstrafing. This is as simple as just holding a strafe key and moving your mouse steadily in the corresponding direction. For example, hold A and move the mouse left. You can switch back and forth to make yourself unpredictable and avoid incoming fire. Experiment with different mouse speeds to see how tight you can turn, but always make the mouse movement smooth and steady. Do not hold W while airstrafing unless you want to do a "helicopter strafe", which is harder to do and extemely situational.
After getting this basics down, you'll want to learn how to jump off of a wall in order to get less height and more distance. It's pretty much the same thing as a standard jump, but instead you fire the rocket at a wall/prop that you're brushing up against. The higher up you aim on the wall, the lower you'll go.
This video shows these basics, with the first jumps done next to the cp_granary mid crates so you have a point of reference as far as height goes. Key press are on the center of the screen, with J being jump and C being crouch.
Notes on Jumps
Just a few notes on jumping:
1. Holding crouch when you're airborne doesn't affect the jump itself in any way, but if you get hit by an airshot you'll catch much more air (provided it hits your leg area), and if you get hit by hitscan you'll lose a lot more momentum. Also, crouching in the air raises your feet, so use it if it'll let you make the landing.
2. Since crouching in the air raises your feet, you have a tiny bit further to travel before you land, which can make you take more fall damage. The increase is pretty small (from some quick testing I'd say in the 1-5 range), but sometimes that little extra health can be the difference between landing alive and cratering.
3. The Original is completely viable for jumping, but keep in mind you'll have to aim further back when jumping to compensate for the rocket coming out the center instead of the right. Of course, if you're jumping off something to your left, you don't have to turn as far as with the default launcher.
Pretty much what it sounds like. You do a standard rocket jump, then fire another rocket at about feet level on whatever wall you happen to be flying past. Like with the standard jump, you'll want to place your rocket a little behind you to get the most distance. Reaching a high enough speed to need to aim more forward only really happens in jump maps. Make sure you're holding crouch during the jump.
These are basically wall jumps, but done off the ground. They let you fly over large distances really fast if you nail them right. To do them, start with a horizontal jump (how horizontal depends on what you're skipping off), and as you fly over the ground (or anything really), fire a rocket down at your feet while holding crouch the whole way. This is done without touching the ground. Also, keep in mind that the faster you go, the more you'll have to aim ahead of you to compensate for your speed.
Speedshots are a little like skip jumps, except you touch the ground. The timing is so precise that it's really hard to get consistent with, but it can be used occasionally. Setup is the same as a skip jump, but hold your rocket until you land, and fire it a split second after landing, right beneath you. If you just go up a little, you fired too late. If you don't really go anywhere, or just a tiny bit forwards, you fired too early. When done right, you can get much more distance than a skip jump.
Ctaps (also known as ducktaps) are jumps where you tap the crouch button, then jump and shoot as you're traveling up out of the crouch. This will give you more distance than a regular rocket jump. I don't know the technical specifics behind it, but as far as I know, it's the raising up out of a crouch that gives you the extra boost (recently I've also heard that it's because your player model actually sinks a little into the ground as you crouch, which means you're closer to the ground when you jump and therefore get more force out of the explosion). Type "cl_showpos 1" in the console, and then walk around tapping crouch for different lengths while keeping an eye on your velocity. You want to tap crouch and let go right before you start to lose speed. Then as you're almost completely raised out of the crouch, jump and shoot at the same time. So the two parts to learn are how long you have to hold crouch in, and when you should jump/shoot. As you probably realize, the timing has to be exact, but once you learn it you'll be able to do jumps such as from gray bridge to the top of the spire fence on Badlands.
Trash to Spire
Useful for quickly getting on spire, but with the spire deck being almost as high as an optimal rocket jump, I would not recommend this jump if there's someone on spire watching you, since it will be incredibly easy to deny the jump. I find this jump to be a little easier when you fire the rocket at the lowest part of the wall.
Patio to Spire
The same idea as the previous jump, but for when you're attacking the enemy spire (or if you're behind the other team). To make this jump easier, you'll want to carefully aim your rocket instead of doing a snap turn. It's definitely one of the hardest jumps, and personally I haven't even mastered it myself, since it should be possible to land on the point directly. Alternatively, you can do a jump off the right wall. I don't think it's possible to land on the point/ledge with this one, but it's easier to pull off and you should land on the last little ramp going up to the point.