If you permanently add an engineer to the room you gain 5+ extra power. Power Relay helps provide power to the Avenger. You’ll want to get the Squad Size upgrades as soon as possible so you can bring up to six soldiers onto the battlefield at once. Squad Size I and Squad Size II allows you to add one extra person to your squad per each Squad Size level.
This includes increasing abilities for Snipers and Grenadiers, as well as being able to upgrade things like Lightning Strike, where you can get additional mobility while being concealed. You can choose skill tactics for a soldier in the Guerrilla Tactics School as well. Guerrilla Tactics School lets you choose a class for a soldier. Additionally, upgrading the Psi Lab allows you to have two soldiers train at a single time. If you staff an engineer in the Psi Lab, you can increase the training speed of a soldier. While training the soldiers you can’t take them into battle until their training is finished.
You must select a rookie to enter the training, not a veteran. Psi Lab is where you train your soldiers for psychic abilities. You can check out the video guide below covering the Avenger facilities and how you can customize, upgrade and build your base.
YouTuber ChristopherOdd has a great series of helpful videos that break down simple things like what each facility does in XCOM 2 and how early or late in the game you’ll want to build them, as well as cover some of the skills and abilities of the classes so you know what best suits your play-style.