In the start i try to research laser rifles, because they are cheap to research/manufacture, and have more accuracy then normal ammo rifles.
Then lots of grenades, and proximity grenades, i found them to be lot of fun. Grenades and proximity grenades are the best to use in the start of game since the soldiers cant hit for shit.
I usual dont use all the time points a soldier have, i try to hide behind building ect, so if a alien steps out infront of my soldier, my soldier have a chance to shoot and kill the alien before it can retalitate. And also use proximity grenades if i know a direction that the aliens come from. Even if the proximity grenades dont kill the alien its morale/health/accuracy is reduced, and i know exactly where the alien is. They might even panick/beserk next round.
If i have landed at a ufo with 2 entrances(ie supply ship/terror ship/ect) , i usual mine 1 of the entrances up with a proximity grenade, and hide one of my soldier around the corner, if the grenade goes off i can usual kill it with my soldier next run, or place a new proximity grenade. Some of this tactics also work inside ufo's, Large scout and up, just dont cut your own pathway with a proximity grenade
Aliens take control over a soldier inside our transport ship:
Well this had happend a few times for me, when i have soldiers i dont currently use, i found out that get the soldiers down on their knees help a good way, it rarely one of them get attacked by a mind attack.