You're browsing the GameFAQs Message Boards as a guest. Sign Up for free (or Log In if you already have an account) to be able to post messages, change how messages are displayed, and view media in posts.
The only two ways to get PA training are the BoS and Arcade's quest For Auld Lang Syne. The former requires you to complete a chain of quests and the latter requires you to be on the verge of the finale.
Well, you can get Powe Armor Training by becoming a BoS Paladin, or completing For Auld Land Syne, just know that you'll probably have to kill a power-armored old man with a Gatlin Laser if you don't have enough speech. Non-Faction Power Armor is rare, I got one from the BoS vendor after becoming a Paladin, and the only other place I know of is Deathclaw Prominotory, and the name can clue you in on what to expect there.
You can begin the quests for the Brotherhood whenever you want. You need either Veronica, which is the easiest way to obtain the quest, or to find the three Brotherhood patrols, which is something you'll have to do regardless, so I recommend it.