If you view the Ascension ability tree, you will notice there is a lot of abilities to unlock some of which cost hundreds of AP. AP doesn’t drop at a steady rate while you play, but it is a resource similar to XP. There are always opportunities to get more everywhere you venture.
If you want to get AP fast and don’t mind grinding, it’s recommended to find a group of low level creatures you are able to quickly kill with Warp Striker over and over to get 1 AP per kill. One great location for this is west Duscae near the Disc of Cauthess, where you can get more than 800 AP per hour a lot of Gils and XP at the same time. You need the monster whistle to call enemies to your locations (which will be obtained up at Chapter 4) and you need to be there when it s not raining, otherwise it is only about simple killing creatures over again.
Having certain AP boosts will increase your AP gains, consider doing some workarounds prior to grinding to maximize the effort. You can get AP from different sources such as taking tours, leveling up, training with Gladio, performing specific kills and abilities during a battle, but if you take your AP gains seriously, consider to invest your first AP points into Exploration part of the menu that gives AP for doing other activities like Roar-running (32 AP) makes you earn AP from driving in the Regalia. Choco-bump will make you get AP from riding your Chocobo.
Then there are options such as getting AP from winning Chocobo races, taking photographs, catching fish, visiting new camp or when you cook a member’s favorite food. It’s recommended to get the Road-running (using the Regalia), Choco-bump (using the Chocobo), Happy Camping (camping in a new location) and Magic Action (getting Ap from casting a spell) first, because these are all activities you will be performing regularly when you start your journey throughout the whole game. There are also AP boosters from Armiger and Magic areas. For example Armiger Action (48 AP) will make you earn AP whenever you use the Armiger ability. Magic Action (24 AP) will make you earn AP whenever you cast a spell.