Probably the most important thing is to always have a backup plan.
Don't trust your soldiers' lives to the dice. If you're taking a shot that doesn't have a 100% chance to hit, be prepared for it to miss. Instead of staying in firefights with the aliens, suppress, retreat to a better position and flank. The heavy with his guaranteed 6 damage rocket launcher is your best backup, so move him last. Grenades are a decent alternative. Also, when moving your soldiers, use the first move on all of them, and only then start taking the second moves. That way you can better react to new enemies that you might find during your turn. And move the person that's the farthest back first, so you won't accidentally move too far and trigger new enemies that some of your guys can't reach.
If you can avoid it, don't end your soldiers' turns in the currently visible enemies' line of sight, even if you get to full cover. For example, don't leave a soldier standing at the corner of a building if an enemy can see him; move him one tile away from the corner so that he's fully behind the wall.
Also remember that when the aliens first spot you, they get a free move, but not a full free turn. New aliens will never fire at you before you get to react. This means that if you haven't yet found any aliens, you don't have to worry about getting into cover. Just focus on moving to a better position, or on setting up overwatch traps. And keep count on how many aliens you've seen and killed, so you won't get surprised.
That's all I can think of for now.
Don't trust your soldiers' lives to the dice. If you're taking a shot that doesn't have a 100% chance to hit, be prepared for it to miss. Instead of staying in firefights with the aliens, suppress, retreat to a better position and flank. The heavy with his guaranteed 6 damage rocket launcher is your best backup, so move him last. Grenades are a decent alternative. Also, when moving your soldiers, use the first move on all of them, and only then start taking the second moves. That way you can better react to new enemies that you might find during your turn. And move the person that's the farthest back first, so you won't accidentally move too far and trigger new enemies that some of your guys can't reach.
If you can avoid it, don't end your soldiers' turns in the currently visible enemies' line of sight, even if you get to full cover. For example, don't leave a soldier standing at the corner of a building if an enemy can see him; move him one tile away from the corner so that he's fully behind the wall.
Also remember that when the aliens first spot you, they get a free move, but not a full free turn. New aliens will never fire at you before you get to react. This means that if you haven't yet found any aliens, you don't have to worry about getting into cover. Just focus on moving to a better position, or on setting up overwatch traps. And keep count on how many aliens you've seen and killed, so you won't get surprised.
That's all I can think of for now.