There are 4 Power Armor Types in Fallout 4: T-45, T-51, T-60, X-01. This video guide shows you locations where you can easily find them. It depends on your character level what armor type you will find! If you are only level 5, odds are you will get T-45 parts (which is the weakest). And if you are level 50 you should be able to find T-60 and X-01 armor. But even then, it is slightly random what armor type you will find in each place.
Most of them already have fusion cores installed. So if you are simply after a fusion core, you can check these locations and extract it from the power armor. Upgrades can be made at Power Armor Stations to further improve the armor rating.

Some of them require an upgraded Hacker perk for Advanced and Expert terminals. You can go to all these places right after the game starts! Please note that there are more Armor locations than this video contains. Even enemies wear them and you can loot it. There are countless ways to obtain new parts, these are just the easiest locations to get you started in the early game.

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