After gaining permission from Elder Lyons, speak to Paladin Gunny in the A-Ring of the Citadel. There is no charge or other requirement for training - the screen simply fades to black as Gunny begins teaching you. Once that's done, you may use any kind of power armor, including the Enclave and the Outcast variants.
If you do not ask for training on your own, you will get it automatically at the end of The American Dream when you become an honorary member of Lyons' Pride, however these options appear very late in the main quest.
The easiest legitimate way to acquire the perk early on is to go through the Operation: Anchorage simulation directly after leaving Vault 101. The distress call will appear within 3–5 minutes of leaving the vault. It allows you to bypass having to speak with Paladin Gunny in the Citadel.
It is possible to obtain the perk without completing any quests past Escape!. There is a mapping bug northwest of the entrance to the Citadel on the Arlington Bridge (The only bridge that fully crosses the Potomac River). There are a few mounds of rubble that you need to get to the top of. The "bald" mound of rubble allows you to fall in, once under the map, work your way to the Citadel by going left when you can. See it in action here.
If McNamara remains Elder, wait 24 hours outside of the bunker for the lockdown to be lifted, then talk to him. Tell him you wish to join the Brotherhood of Steel and this will begin the Eyesight to the Blind quest. When this is completed you will then receive Power Armor Training.
If you first complete For Auld Lang Syne and later complete Still in the Dark, Elder McNamara will be astonished that you already had Power Armor Training and will ask where you learned it from. However he then brushes it off as a result of the Courier being "well traveled" and will not give you the option to alert him of the former Enclave bunker nearby.
The player is the only person in the entire game who needs a special perk to wear power armor, companions like Raul, who would likely have no experience with power armor, have the ability to wear it. Even raiders can wear power armor.
If you complete all of the preliminary quests for Elder McNamara, and also help have him overthrown it is possible to end up with no way of receiving Power Armor Training, as the Elder will tell you he is aware of the plot to over throw him and to return later, and Hardin will also ask you to return later as he is busy. It doesn't seem to matter how long you wait to return if this happens.
In Honest Hearts, there exists a dialog option enabled by this perk when talking to Ricky, where you can refute his suspicious claim that he shot a Brotherhood of Steel paladin through the eyeslit with a "11mm sub machine gun", by stating that the eye slits of the T-series are bulletproof.
Through normal gameplay, training is required to wear power armor of any variety. However, the skill can be unlocked via the console in the PC version by typing player.addperk 00058fdf. An alternate way to do this is to type SetPCCanUsePowerArmor 1 in the console.