What's a good Let's Play video that has a good overview of the systems of this game?
Most of the streams out there have campaign spoilers. Paradox put out some good ones on their YT channel.
The campaign involves running the day to day of your mercenary group:
You 'advance' by salvaging and/or buying better equipment, training your pilots, and raising both your mercenary rating and your relations with the various factions of the game. You can upgrade your dropship as well. Things like travel and repairs take time, meaning you must watch your cash reserves to ensure you have enough to get by.
A significant amount of time will be spent tinkering with your mech loadouts:
Mechs come in different weight classes, with each chassis having its own unique flavor. Weapons are a function of what you have in inventory and what type of hardpoints you have available on your mechs. You can put mechs you don't want in storage or simply sell them to try and keep a healthy cash reserve - you need to pay for the day to day after all.
The combat itself is very familiar to Xcom players, with the nuances of Battletech, like crits, ammo explosions, armor facing etc...
(reached my media limit):
https://www.youtube.com/watch?v=mDQIQ6-L_i0
The game should be easy to pick up and offer quite a bit of depth for newbies and veterans alike. If you are into more chill turn based games, this would be right up your ally.