I've played soldier for around 1000 hours on my account, and have played it on several different teams in the competitive environment. In this post, I'm not going to discuss the basics of the class; instead, I'm going to focus on the little tips that I've learned over the past two years.
First and foremost: the default rocket launcher + shotgun loadout is hands down the best if you are looking to be an effective part of the team. Obviously, no melee can beat the equalizer, so it wins over the shovel.
Why the rocket launcher (RL) over the black box (BB)? Yes, the BB does give you extra health on hits, so it allows you to win most 1v1 situations with relative ease. However, the way I play, I enjoy having a medic healing me most of the time (this is known as "pocketing" the soldier, in competitive lingo). It lets you be a bit more aggressive, and opens up your options a whole lot more. In that case, having all four rockets is absolutely necessary, as you find yourself in situations where you have to do large amounts of damage quite regularly. With the BB, you lose clipsize, with a fairly redundant health regen feature.
Obviously, in a public server with no medics, the BB may be your only option to stay alive. However, using it is like having a handicap--you can make lots of mistakes with your aim, allowing your enemy to stay alive longer, and still come out alive in situations.
You want to deal as much damage as you can, as quickly as possible, while ensuring that you have enough ammo left over to address any other threats immediately.
I often see players waste all four rockets in the chamber, and then immediately fire any reloaded rockets, because they don't feel confident enough in their shotgun aim. Improving your shotgun aim makes you twice as powerful of a player, because you can defeat much more versatile classes, such as the scout.
If you are playing soldier, try to limit the number of rockets it takes to kill enemy classes. A good rule of thumb is this: for light classes, such as scouts, snipers, or medics, use one rocket only, then finish him off with a shotgun blast. For heavier classes, such as soldiers or demomen, use two rockets, then switch to shotgun. For heavies, who are often overhealed or attached to medics, do whatever works. By forcing yourself to use both weapons, you will conserve ammo, and leave yourself with more options when the next enemy appears.
Your goal when aiming your rockets is to "bounce" your enemy into the air, where they have less control over their motion, and are thus easy targets. Aiming at the ground immediately behind the enemy will often produce the desired effect, as the splash damage will propel them upwards, where they can be finished by a rocket or shotgun blast. As far as airshots go--they look very flashy, but are also easy to miss. Your shotgun is much better at finishing off airborne targets. Use it!
If you are being healed by a medic, and you notice that his/her health suddenly drops sharply, or he calls out an enemy that you cannot see, immediately turn and blindly fire a rocket directly onto your medic. This will ensure that you blast any classes assaulting your medic away from you, and gives you a little more breathing room.
On rocket jumping: knowing how and when to rocket jump will make you an extremely deadly force. Most players do not fully understand how to utilize the speed that good rocket jumping can provide. It allows you to mount surprise attacks on the enemy, or escape from a bad situation, and will always give you a height advantage on your opponent, all at the cost of around forty to sixty health.
http://www.youtube.com/watch?v=xhaTX5pP274#t=0m56s is a great example of this. The rocket jump is just short enough to land the soldier on the pipe, and doesn't take too much health. Without jumping, he would have been at a height disadvantage, and probably would have been killed.
Playing without a crosshair will make you better. I can't explain why, but after you spend a few weeks adjusting (the shots that require precision aim will, I admit, be a pain at first) your aim will be much better. In addition, "flick shots" (ones where you don't consciously manually adjust your aim, but instead trust your reflexes) give you better, if slightly less consistent aim, provided you've played the game long enough that you have muscle memory.
Hope some of these help, and as always, feedback is much appreciated!