Ascension is the main way to power up your characters, outside of equipment and leveling them up. It has similarities to the Sphere Grid from Final Fantasy X, as well as the Crystarium from Final Fantasy XIII, although it is not a single entity as with the former options. Instead, the Ascension option is split into many different "boards," called Nexuses. Each nexus is associated with a specific aspect of the game, like magic, exploration, and many more. The majority of things are already unlocked from the beginning, but some more nexuses will become unlocked as you progress throughout the main story.
In order to purchase said nodes on the Ascension board, you will need to earn AP while playing. One of the easier methods is through combat, but only specific actions will grant AP. These are by defeating enemies with a Blindside Link, Linkstrike, Parry and Warp Strike. Other methods of obtaining AP include some of the choices when presented options during conversations, and the Tours sidequests you can get from your buddies when camping at certain spots. Also, some of the nodes you can buy from the Exploration section of the Ascension board will give you the ability to earn AP while doing some of the tasks you'd normally do during gameplay, such as fishing, camping and even taking long trips in your car.
So, you're wondering what exactly is good to purchase in the Ascension trees? Well, look no further than the following suggestions. To start off, it's a good idea to go straight for the Exploration nexus, as it has nodes that will grant you extra AP for doing mundane stuff, like driving the Regalia (Roadrunning), camping (Happy/Happier Camping), riding Chocobos (Chocobump) and even fishing (Angler Action). Once you get a lot of recipes for Ignis, look into the Aftertaste and Lingering Aftertaste nodes, as they will increase the duration of the buffs from meals.
On the Stats nexus, you definitely want the second accessory slots for each character, but the third slots might be a bit expensive for the main game (definitely want them postgame). If you use Magic, look into Elementalism to draw more energy from deposits, as well as Powercraft and Extra Powercraft to increase the potency. Hang In and First Aid are excellent choices on the Recovery nexus, as are Armiger Harvester and Iron Armiger in the Armiger tree. For Techniques, Prompto and Gladio are pretty good with their default ones, but Ignis has some very powerful techniques in Regroup, Enhancement and Overwhelm; don't forget Tech Strike to help increase the tech bar during combat.
The last two trees are Teamwork and Combat. You will want to purchase the nodes that increase damage for Blindside attacks, Warpstrikes and Linkstrikes in their respective nexuses. Also, Analyze is great for Ignis to help out with identifying weaknesses, especially if you're not using Wait Mode. Venom Fang (Ignis), First Strike (Prompto) and Reflex (Gladio) are sound choices, since they are relatively cheap. Light Phase helps preserve MP for Warpstrikes by reducing the cost when dodging. Airstep will give more maneuverability in the air, while Airslip will allow you to dodge while in the air.