Most things have been said, so let me just add a couple of things:
- if you haven't triggered any aliens, you will not be shot on your turn even if you trigger aliens, and if you run into a patrol, they won't shoot you before the next turn after that. You can, and should, abuse this mechanic like shit, because it means that when you're out of combat, you can.. ah, screw it, just watch Beaglerush's videos for an explanation [http://www.youtube.com/watch?v=bh26HyUe3po], as well as overall tips on how to do things.
- Avoid prolonged firefights. Ideally you don't want any of your crew members to get shot at, because that's a risk that the gods of RNG decide to kill you with a bad dice roll. If you need to draw fire or bait some aliens in, try to bait with a squadmember hunkering down in heavy cover, as that makes him/her almost impossible to hit. Some aliens will waste a turn on this, others will just not take the shot and move around, hopefully to a more vulnerable position. When you're free to move in and shoot them, do so.
- For most of your crew, you're going to want them completely out of line-of-sight if possible, as it means they won't get shot and are far more free to try flanking maneuvers. This isn't always possible, so stick to heavy cover in all other cases. Also, unless you have a smoke grenade or use hunker down, don't use light cover. It's a measly 20% hit reduction chance, whereas heavy cover is a 40% reduction, and sight-blocking cover means you can't get shot at all.
- Grenades are lifesavers early on if you use them right. A single grenade will, on classic, insta-kill all sectoids in its blast radius. This means that, as long as you have someone with a grenade within throwing range of the sectoids and don't trigger a new pack of aliens on your turn, you can let everyone take bad percentage-shots on the sectoids in an attempt to kill them, or even move into a flank where they themselves would be out of cover, and then take a shot at the sectoids. The reason for this is simple: if your guys hit their shots, you don't have to use the grenade, but even if they miss their shots, you're guaranteed to kill the sectoids in question anyways, so you're not taking any risks at all.
The first three months of the game are, without any doubt, the hardest. If you choose the right abilities for your soldiers, don't let panic get too high and don't lose your entire high-level squad for some reason, you'll probably win. As for the abilities, I don't necessarily agree with the builds posted in the guide in the first reply here(Suppression is not a "situational ability", it's a goddamn useful all-purpose ability), but they're functional at the very least.
