The best advice I could give would be to take your time with your turns and trust in the probability that there is a solution for each turn. I don't know if it's an AI trick or something, but even on Hard difficulty, there are very, very few turns where you logically can't avoid losing a building. It's just a matter of finding it, similar to those old "checkmate in 3" puzzles in the newspapers.

Also a basic tip: a good way to start a turn is to identify the "critical" enemies: the ones that threaten the buildings. The rest can often be ignored (not really, but we're talking basic). Most of the time you'll have 3 mech actions, and 3 critical enemies, and you just have to tie those together. Later on, you will find yourself in situations where there are more than 3 critical enemies, and that's where you'll have to pull the real shenanigans.
Thanks for the tips! I can definitely feel myself getting better. And I typically realize that it's my own fault when I paint myself into a corner and end up in a situation where I have to make hard decisions about what to sacrifice. But it's also immensely satisfying when you manage to pull off some crazy trick where all the enemies end up killing or obstructing each other.

Unlocked a third squad today, the electric storm ones (perhaps my favourites so far, it's nice that the storms are both a DoT effect and a temporary lockdown). I like how they all seem to have different playstyles. But I guess that makes sense from the designers of FTL, where it was also great fun to try out the different ships with their strengths and weaknesses.