Mana is for attacks in battle. If you hover over the attack buttons, it'll tell you how much mana they create or spend(some require a certain amount of mana before they can be used).

Experience is gained by defeating the enemies in a battle. The higher the level of the enemies is than your pets, the more experience you gain; the lower, the less. (Ex, if your pet is lvl 25, a lvl 30 enemy would give a lot of experience, but a level 20 enemy would drop very little)

Also, missions are available in Barracks in towns. Your pet has to be the proper level for the mission, and their level up/proficiency points must be best in the stat needed for the mission to succeed.

You can eventually unlock more mission slots from the person below the Healer in each Plaza, for up to 6 slots. A lot of users raise pets specifically for missions in order to have more success with them and to not go without their battle team.

Quests are acquired while proceeding through the storyline and at Quest Boards.

Thanks!
What are proficiency points for?

They further enhance stats, and every 5 points you get a new attack based on the proficiency tree you chose! Seeing as you get a point every 5 levels, you get an attack at level 25 and other at level 50.