I think a lot matters as to what difficulty level you're playing on. I know in the harder difficulties, the first few missions are a real dog. You have to pick and choose to some extent, although the consequences of failing a mission are pretty bad. But there's almost no way you can survive a night terror mission at the beginning of the game. Your soldiers will all be dead before you even see the first alien. Researching plasma weapons is the key at the start though, along with alien grenades which are both plentiful and do great damage. If you aren't packing heavy firepower by the time the disks show up, forget it. In the early missions, rely heavily on your tanks for scouting and killing. Your soldiers can use your tank for cover too. And most importantly, remember that if any member in your squad can see an alien any other member can shoot at them if the have line-of-sight. If you have your troopers on an elevated position and a tank out in front spotting, your soldiers can take pot-shots from long range without much fear. This is critical to winning the game. If one of your soldiers spots an alien, do NOT use him to shoot at first. Use those soldiers following behind to use aimed shots and snipe the alien. If the spotter shoots, then the alien can shoot back. If you use snipers, the alien will not. Done well, you can use your tank to spot out aliens, snipe them with your soldiers and still have TUs left to pull your tank back to cover.
Once Mind Control starts up, ditch any soldier that is effected by it You won't be able to see your soldier's ability until after you have mind control labs, but if they get taken over in a single hit, you can bet their score is pretty lousy. It sucks to find out that your best soldier is mind control weak and you have to ditch him, but it's better than having him wipe out your whole squad.
I forget what you need to research to get alien armor but I remember it being difficult to find. Once you get decent armor and mind control for your own soldiers, the game becomes significantly easier, even on the hardest difficulty levels.