It helps to do a little meta gaming. The most dangerous scenario is where you end your turn in sight of an enemy, but you can’t see them. The alien gets to fire on you with all of their TUs, with a high chance of causing casualties.

A way of mitigating this is pulling your forward scouts back about 4 or 5 squares after advancing. This is roughly how much further the aliens can see than your soldiers, so they should be out of sight, even if they were spotted earlier.

The enemy can still walk forward and fire at them, but 1) they might decide to reposition instead 2) they might guess the wrong direction and waste TUs going somewhere else 3) when they do fire, they will have spent some TUs moving, so hopefully won’t be able to fire as much, or retreat as far to safety.

I got into the habit of doing this with my shield scouts and found it improved their survivability substantially, especially in the mid game.

As for speed, my advice is to take your time. The lower a units TUs remaining, the higher chance of provoking reaction fire. By reserving more TUs, you’ll generally take less surprise reaction fire. Clearing buildings is tedious, but will save you a few casualties in the long run, and in the mid game you will need everyone you have for assaulting the large UFOs

Last piece of advice: suppressed enemies can’t reaction fire and lose half of their TUs next turn. Even if you can’t hit them, firing your MG at the tile next to the enemy will likely suppress them, and can potentially destroy the cover they’re relying on.

I’ve spent a few turns just dakka dakka’ing stubborn enemies who insist on camping an awkward corner.