I destroyed this game on Normal but am getting my ass handed to me on Classic Ironman. I've had 5 games in a row not get passed
I guess I need to work more on flanking or something...
Flanking is actually dangerous on classic ironman because it can lead to finding even more aliens, which can make things even dicier. Rather, I think in the early game the most important things are what milkham has said, and use explosives. In normal you probably never really had to think about using them because flanking was much easier done without triggering too many extra mobs ( I think there is a limit of 5 enemies you can encounter at any given time in normal), there is no limit in classic and above, so you must be alot more careful about the number of aliens you reveal or you risk getting swamped. Thus I think it is alot better to work on knowing when to use the defensive abilities and explosives than to worry about flanking.
If you see a few aliens behind cover and you can't easily flank them without revealing a huge portion of the map, toss a grenade at one of them and try to focus down the others. If ever you feel like there are too many aliens for your squad to handle, blow up a few of them. If there's an alien that none of your squad has a good shot at (or half the squad has missed), just throw a grenade at it, if it doesn't die the rest of your soldiers should have a much better shot at it (or can kill it with more grenades).
I think the game does a disservice to those playing the higher difficulties by having Dr. Vahlen so directly telling you not to blow aliens up. Don't be afraid to kill aliens with explosives and lose weapon fragments in the early game, you won't usually need too many of them until much later. As long as you don't kill more than half the aliens with explosives in any map, you should have plenty of weapon fragments to work with until you reach plasma weaponry, at which point most of the aliens you face won't die quickly to explosives anyways, so you'll be using explosives more to remove cover and open up line of sight rather than to kill.
Whatever you do, in the early game make sure at least one person has explosives to deal with cover and line of sight. Its debatable whether its necessary in the late game, but in the early game you don't have very many options in dealing with the aliens besides just shooting, overwatching and hunkering down so being able to blow up cover or guarantee a kill can be extremely helpful.