Espers can be enhanced just like the actual characters, but they do not level up by consuming each other, like the heroes do. They gain levels when they are equipped in battle, of course, but the enhance option allows them to gain quick levels, in exchange for consuming asper-enhancing material. You will want to try to match elements, as well, so you can get the 1.5x exp bonus and make the most out of an element.
Training an esper allows you to fill out the training board using the SP that have been gained from leveling up. Spend them as you want to on the stat boosts that you see on the training board. Once you pick one stat boosts, the connected stat boosts will become available. It’s similar to the Final Fantasy X leveling system, in a way.
Once an esper reaches its maximum level, it can then be evolved. To evolve an esper, use it to fight a battle against the same kind of esper, and then if you win, the esper will evolve. Once it does evolve, the size of its training board will increase, as will its stats and its summon magic.
Add friends and respond to your friend requests. Not only can you add their leaders (and they can add your leaders) in battle, but you and your friends can send gifts back and forth. Go to your friend’s list as often as possible to send and receive gifts. Set your desired gift, and auto-send to your friends to automatically send them their desired gifts.
Complete trophies in order to earn free Gil, free Lapis, and materials and crafting items, and even free characters. Go to the Trophies area and tap on any one of the darkened trophy requirement icons to see what you need to do. Then go to the Rewards area to claim your rewards, if you have any to claim. If not, then go out into the game and complete a trophy.
The chains can seem impossible to pull off in battle without hitting auto-battle, but they are both possible and doable. Attack with multiple attacks of the same element in a row to pull off an element chain. To do a spark chain, attack with multiple attacks at the same time. The function of a chain is to increase the damage done by the set of attacks. This, of course, will require a lot of fast tapping.