From what I remember, soldiers with best starting psi skills where soldiers with Spanish names (not sure if it was restricted to soldiers from South America). So these guys are also quite immune to or have a lower chance of being mind controlled right from the beginning.

You could thus go and dismiss every soldier not having a Spanish name and keep the others but at the beginning of the game, this would be too expensive for the money you have.

Every soldier using up time units for basic actions at least once per mission will get higher TUs for next mission. In fact I think it is only the first basic action that triggers it. The more actions you do has no effect on how many TUs you get.

Whenever possible make your soldiers use up as many of their TUs running around. Their stamina will get much higher for the next mission.

Well this one is a bit tricky, but every time your soldiers get hit and healed at the base their health bar increases.
What you can do is shot the trainees around the end of the mission with normal pistols as they deal less damage and the risk of killing them is lower. The next mission they can participate on they will have a much higher health.

Every time morale of a soldier drops below 50 and he does not panick, he will gain more bravery at the end of the mission.

Every time a soldier reacted to an enemy (e.g. reaction shot) he will gain more reaction points at the end of the mission.

The more often your soldiers hit, the better their accuracy skills get. You can thus get 100% hit chances on automatic shot by 'training' your soldiers.
If you have enough TUs to get close to an enemy not looking into your direction AND still be able to use automatic shot: get behind him and pump him full of bullets. In later missions I tend to give the trainees I take on missions normal rifles or pistols with plenty of ammo as they do less damage so they can shoot more often. As I said EVERY single hit makes them better.

A good strategy is to make a better (armoured) soldier trigger a reaction shot from the enemy so that the enemy has no TUs left. The trainee gets as close as possible and starts shooting. But make sure you can then either kill the enemy before the end of your turn with another soldier or MC him, so that he will not kill the trainee on next enemy's turn.

Same as for shooting. The more you use it successfully the better you get.

To get soldiers that can carry whatever you want without losing stamina too fast. Cram your trainees inventory with as many items you can. They will be stronger next mission.

As soon as you get MC use it as much as possible. Every use of it makes your psi skills get higher for the next fight.