Hi Didact04, I'm on my first vanilla playthrough as well.
My tips so far for the early game are: a hospital or 2, a big team of rotating squaddies, rocketeers and some guys with shields as point men/ tanks/ observers, shotgunners and a sniper. Basically anything but the riflemen with ballistic weapons. It's a slow methodical advance, rocketing cover at a few, select points. Then rocket the UFO front door and lob some gas grenades in if there's more than one alien in the entrance. If you're having a problem with cover, then smoke grenades can help protect your team. I don't like the normal grenades, they seem to take too long for too little effect (a run up and shotgun is usually much better).
At one point in my early game I had 2 hospitals full of wounded soldiers, I think I was lucky that few died, but some were out of action for ages.
Steel Monkey