The strategy I use in general for Insane difficulty is:
3 shields to search the map.
One Rocket Launcher to take down cover and remove buildings.
The Shields go first, because they're the least likely to trigger reaction fire and the most able to tank it. The aliens in X:CE will target non-shielded personnel over shielded personnel given the same chances of hitting, so putting your shields kneeling down out in the open or standing up while the rest of the team stays kneeling or in cover will allow your shields to tank fire for the rest of the team and prevent your squishier personnel from being gunned down.
Once the scouts find someone, the 2 MGs, 2 snipers, and Rocket Launcher alpha strike them with all their firepower. Hopefully, that kills them before they can return fire, but if not the shields should hopefully be able to take it.
For clearing UFOs, you want to make sure that everybody has the maximum amount of TUs possible before entering a room and that your shields go in first.
Make sure to save before you start the breach if possible.
For Landing Ship UFOs in particular, once I've cleared the bottom floor I generally split the squad into 4-man fireteams, with one shield, sniper, and machine gunner attached to each team. Typically, I get the guy with the rocket launcher to drop it and pick up an alien weapon or a spare shotgun. Approach the teleporters cautiously with plenty of TUs in reserve. Get each team to surround the teleporters in a semi-circle so that nobody accidentally shoots eachother with reaction fire.
If you're on CE, don't stand on the teleporter. The aliens can telefrag you. The shield guy should be facing the right direction to absorb reaction fire when he goes up. The Machine Gunner should be facing the direction that will take the least TUs to manuever him into position. The rest should be next to them, right next to but not ontop of the teleporter.
End turn, so that you go in with max TUs.
Send the Shield guy in first, have him turn and check the entire first section of the upper floor to make sure nobody's waiting in ambush.
If there is, kill them, either with the shield guy or by bringing up other people and teleport back down.
End turn and then send in each team one by one to line up by either door of the upper floor. The shield guys go in first and kneel in front of the door. The MGs go in next and get in the right angle to fire, typically standing right behind the shield or off to the side behind the shield guy so they can fire over him. If you have experienced MG guys with around 90 TUs, if you position things right you can have them open fire on the first turn you breach the upper floor. The snipers and secondary shield guy/former rocket launcher guy go in next and kneel to either side of the doors, turning so that when the door is opened they can see aliens on either side, although there's a blank spot in the very corner.
One of them, whoever is least likely to draw fire, opens the door.
(Whoever has the most TUs remaining, has a reflex weapon like a shotgun or pistol, or is out of LOS due to being off to the side of the door are the soldiers least likely to trigger reaction fire. Failing that, whoever has a shield and can hopefully tank it.)
The machine gunners open fire to suppress or destroy, the snipers take potshots, the rest chuck in grenades, and somebody reserves enough TUs to actually close the door so the aliens don't return fire when the turn ends. This is usually about 12-24 TUs depending on the type of door.
With good luck and enough firepower, you'll be able to kill or suppress all of the aliens enough so that after you close the doors so they can't return fire, the aliens won't be able to walk up to the doors, open them up, and shoot your men point blank.
The shortcomings of this strategy is that you can't actually see into the corners from the doorway. You can still throw stun grenades and EMPs into the corner to check for and suppress potential aliens if you still have the TUs, or if the aliens are in the very back of the room, already suppressed, or are of a low enough rank and thus not aggressive enough to run up to the door and shoot at you.
The benefits of this strategy lie in wiping out most of the aliens in one decisive strike while crippling their ability to return fire on the next turn. Their ability to return fire is impeded further by closing the doors, which forces them to walk up and open them, draining TU that could be used to shoot your men several times instead of just once or twice.
This alpha-strike method minimizes the amount of fire you end up taking, while maximizing the amount of firepower you can deliver.
Ideally, you want to keep your squad close together at all times with clear lines of fire, so that you can kill your enemies on the first turn you see them. UFO breaches operate on more or less the same principle.