Try to make sure your mans stick close enough at all times, that at least 2 will get hit if something throws a grenade your way. I know how that sounds, but grenades are totally survivable. Plasma not so much.
If you can’t bring to bear every weapon you have by doing so, don’t try to move into range. Back off instead. 90% of survival in XCOM is god-like patience.
Always move into full cover, and if you can’t fire and kill everything in range, hunker down. Hunkering down behind full cover isn’t remotely safe, but if you plan on spending a turn in something’s line of fire, nothing less has any meaningful impact on whether you eat plasma & die or not.
Try to memorise what terrain can’t be destroyed, and prioritise using it (pillars, pretty much).
When you need to kill something make sure all your mans can shoot at it, so position them behind cover & with a shooting action left, before you actually start shooting, overwatching & whatnot. Because chances are 2/3rds of your team will miss, and that you’ll need at least 1/3rd of them to kill a single alien.
Never underestimate your own grenades. Not that you should research improved ones, that’s a waste of time and resources. But your basic grenades can remove quite large areas of cover. You should always try to be careful not to kill anything with grenades unless you really have to. Killing stuff with grenades destroys the loot. Just blowing the crap out of them & then shooting them dead doesn’t destroy the loot.
As soon as you have 1 assault with… Lightning Reflexes? The thingy that denies the first hostile overwatch shot, uhm… As soon as you have that, try never to end your turn without someone in overwatch. Overwatch makes aliens stay put 99% of the time, even if it’s a sniper with a basic pistol surrounded by a mob of mutons. But it also makes them very likely to go into overwatch themselves, so do get into the habit of moving your assaults first. Because unlike XCOM solders, aliens are crack shots on overwatch.
But uhm… Point is that overwatch is how you go about flanking & ganking without having to chase down your targets and inevitably popping every last spawn on the map in the process.
If you’re ever tempted to make a trooper run, save your game, shut it down & go do something else. If you still feel the urge when you come back, ask yourself if the trooper is running towards enemies, into unrevealed map, or into revealed map and away from enemies. Unless the answers are no, no & yes, respectively, what you’re feeling is almost certainly the urge to fail hard.
Popped aliens who can’t see you, generally don’t really do much. The sniper’s radar thingy conveniently lasts two turns, meaning you can spot a gaggle and if they’re not the teleporting kind, you can shoot one of then the next turn to pop the spawn. After which they’ll sit around where you found them, doing nothing. Until you wander into their line of sight.
Beware, if you pop a teleporting spawn without neither of you having conventional LoS, it won’t necessarily stop teleporting. Ever. Usually they do stop at some point, but chances are you’ll manage to kill them or get killed by them before they stop zipping about. If this happens, try to set up around one of the teleport points. The points are fixed, so if you see a spawn materialise somewhere, they’ll eventually come back to that point. Also be aware that if, whether by map glitch or sniper, your first kill is from a teleporting spawn that has not been popped, then every teleporting spawn on that map will materialise somewhere within 20’ish tiles of one of your troopers over the next 1-2 turns (except the one you got your first kill from, that spawn will keep teleporting around as described above). Usually just 1 turn. If you know it’s going to happen, it can be a very quick and safe way to get through most the enemies (bring heavies). If you don’t, you’ll probably get the entire team killed in a single turn. I’m not sure whether this is a bug or not (I played a month or two ago and it wasn’t fixed then), but it’s actually kind of practical once you know about it.