Final Fantasy Type-0 HD is not a traditional RPG, where you start the game and simply follow a set storyline. Instead, it is mission-based, with a central hub that you return to after every mission. In this hub, you can do extra side missions or talk to characters to sometimes earn items, as well as train your characters, buy/sell equipment and much more.
However, you don't have all the time in the world to do this, as the game has a time-limit mechanic in between missions. When you're in Akademeia, you have free time to do whatever you want, such as side missions, talking to certain people and even going out to the overworld. The bad thing is that each thing you do takes up some of that free time and when it runs out, you will be forced to proceed to the next story mission.
As already mentioned, during Free Time, you can do a variety of things, such as talk to NPCs, perform Tasks (sidequests), and even go outside onto the overworld and explore. Delving deeper into the main story will offer you more choices during Free Time, with most actions taking up some of that free time. Should that free time reach zero, then you'll be forced to the next story mission, but you can also talk to an NPC in Central Command to start the next mission immediately should you tire of doing side quests.