In Albion Online leveling is done in the form of increasing the Mastery and Specializations in the Destiny Board through acts that award Fame. Fame is the equivalent of "experience" in traditional MMOs. The process of "Power Leveling" is focusing on getting these fame points as quickly as possible or as easily as possible. Additionally in Albion Online "Power Leveling" is referred to as "Fame Farming" though the latter term is mostly used in correlation to combat fame.
Getting the most fame in the shortest amount of time will increase masteries and specializations which in turn power up your character though increasing item power (combat), increase Resource return rate (crafting and refining), or resource yield through better tiered gear (gathering).
For skilling/training that is not solely focused on fastest fame, and rather a more casual, "Jack-of-all Trades" type vibe. See: Casual Skilling
Combat fame is awarded by defeating Mobs and Creatures in the Overworld, Dungeons, Expeditions, Hellgates, and also defeating players in PvP. Depending on the tier of the enemy mob or creature different amounts of fame are awarded. Higher tier enemies give more fame than lower tiers. In addition, the location of the mob or creature also determines fame. For example in Solo Randomized Dungeons designed for a single player, the fame awarded on a similar mob is less than the same kind of mob in Group Randomized Dungeons which are designed for multiple players in a party, and therefore tougher to defeat. Generally speaking, the tougher, and more risky the method is, the more rewarding it tends to be.
Once you have chosen your method of combat fame training, you must choose a weapon. Generally speaking weapons best suited for PvE fame farming tend to have large area of effect damage to clear multiple enemies faster. Its also worth noting, that weapons not particularly well suited for farming may be leveled through Auto-Respec mechanics more efficiently at the cost of Silver.
Gathering is one of the few areas on the Destiny Board that a player cannot buy their way into as the fame is awarded from venturing out to gather the resource in question. In general, the best way is to go by the most optimal farming for your chosen gathering profession. For more information on optimal gathering see the "optimal" section in Gathering. Other tactics worthwhile are employing other players for escorting you on adventures such as Aspect hunting or venturing in the Roads of Avalon.
Once you gather, the next step is the either sell the resources of your labor, or move on to refining and crafting.
To quickly obtain resources without gathering, see: Trading and buy them from the market. This makes refining one of the fastest nodes on the destiny board to train up assuming you have enough silver.
Once you have acquired the resources. You simply need to refine them to create the materials. For more information on how to do this effectively in the most optimal way, see the "optimized" section in Refining. In general, premium is needed to fully take advantage due to Focus and Resource return rate. Other than that make sure you follow the refining page's notes on refining in the proper City.
To quickly obtain materials without refining them, see: Trading and buy them from the market. This makes crafting on par with refining for speed of training on the destiny board.
Once you have the materials, you can craft many different items from tools, armor, weapons and even houses. It is usually best starting out to focus on one specialization. For more tips on this see: Crafting which goes into more details on return rates and gear setups.