There are three official methods that can grant you the Power Armor Training Perk and one unofficial way if you want access to the armor very early and don’t mind messing around out of the bounds of the game.

Here is some info on Power Armor:

• Power Armor is very heavy. Due to the weight system, Power Armor is recommended for builds that have high Strength otherwise you will often be over encumbered unless you carefully manage your inventory.
• Repairing the Armor requires duplicate Armors. The repair system requires that you use the same armor of better quality to repair your current one. So be prepared to have some spare sets stored away.
• Repairing with similar Armor has caveats. T45 Power Armor can be repaired using T-51b parts, but the reverse is not possible. 
• Power Armor is rarely sold by merchants. This makes scavenging off of enemies the only reliable method to get more sets.

Methods to Get the Perk

There are three main ways to get the perk and one non-standard way.

Note: Spoiler alert for some of the main questline.

“Picking up the Trail” Quest

During the main quest “Picking up the Trail”, the player can speak to Elder Lyons and request permission for Power Armor training. He will then tell you to speak to Paladin Gunny to manage your training and grant you the Perk. This is optional, and is not mandatory for quest completion.

“The American Dream” Quest

After completing “The American Dream” main quest, Sarah Lyons will make you an Honorary Lyons’s Pride, and automatically reward you with the Perk as well as the option of a full set of Power Armor or a full set of Recon Armor, both in perfect condition.

Operation: Anchorage DLC

After completing the “Operation: Anchorage” DLC, you will be automatically given the perk if you don’t already have it. You will also obtain access to an armory in the Outcast Outpost, containing unique armors and weapons, which includes the Winterized T-51b Power Armor set.

The Out of Bounds Method

Normally, The Citadel (where the player can request the perk from Elder Lyons) is only accessible after “Picking up the Trail” which is very deep in the main story quest line. Before that, the gates are shut without any normal way to open it. By performing a very specific out-of-bounds glitch, you can access The Citadel right after leaving the Vault! Be sure to and save before attempting this so you don’t get soft locked out of bounds.

Northwest of the main entrance, near Arlington Bridge, there’s a small mount of rubble that the player can jump on top off. You will see a somewhat bald part of the rubble and by reaching its top, you will fall under the map, and take a small amount of fall damage. Now work your way to The Citadel, going left as much as possible. This will allow you to unlock The Citadel fast travel earlier than you should be able to.

Power Armor Locations

One of the most common ways to find armor pieces is looting dead Brotherhood Soldiers scattered across the Capital Wasteland since they’re always wearing the armor. Here are some possible locations that the player can go right after acquiring the perk

Washington Monument

Going to the Washington Monument, near the bunker entrance, will reveal a couple of dead Paladins wearing full sets of Power Armor in decent condition.

GNR Building Plaza

Outside the GNR Building Plaza (the player will visit this location in the main quest) after the super mutant attack event, some dead paladins can be spotted wearing full sets of Power Armor.

Fort Independence

By going to Fort Independence, the player can acquire full sets of Outcast Power Armor (same as normal Power Armor, but with red and black color scheme).

Following in his Footsteps Quest

During the main quest “Following in his Footsteps”, right after meeting Sarah Lyons, follow her until you find the body of Initiate Jennings lying in a mattress wearing a full set. The armor will be in very poor conditions, so it’s really only usable if you plan on using other parts to repair it and making it your main armor.