The computer cheats. I'm not sure how much, but I'm fairly sure they get stat adjustments as the game moves on to stay dangerous. That might just be my gaming superstition kicking in though

Believe it or not, smoke grenades can be lifesavers. Bring a few with you on every mission. Mutons can shoot about a bajillion times in one turn, and you need every little edge you can get. If you have to move across an open field, smoke is the only cover you're going to get.
As for moving about the map, mid-sized groups are the best. I maintain at least two groups of four to six (or one group of eight to ten for huge UFO's) with a crack shot and some expendable rookies in each. Stack up your squads around corners and end the turn before going around. That way, you have full TU's with which to react to whatever might be on the other side. You will suffer a LOT of casualties in the mid-game, so always keep a steady reserve of rookies at the base and in the Skyranger.
Make sure your teams always cover each other. Have a rookie check down every alley before ending your turn, but be careful to duck them back behind cover.
Laser weapons and higher can destroy most non-UFO walls in a couple of hits. This makes getting around the map much quicker, and you might even get some free kills by uncovering hidden aliens.