At the beginning of Chapter 2, you will be put at the Peristylium, which is the base for the game. For now, just walk around and explore the fountain area until a cutscene occurs, after which, you will be able to explore more of it. Once you regain control, head towards the fountain and you'll be stopped by a girl named Aki, who says to go and meet with your CO.
You can't explore anything outside yet, so head straight and inside of the building. In fact, you need to go straight to Classroom Zero, which is straight on and to the left from the entrance. There, you will meet your new CO, Kurasame, as well as two new party members, Machina and Rem.
Note: If you feel like you're lost inside the building, simply go to the game's pause screen and you'll see a map there. The areas with the blue lines are accessible via the warp in the central area, while the rest are through doors.
After the introductions, talk to the moogle, who Cinque decides to name Moglin, as well as the girl standing near the front of the room, who introduces herself as Aria; she will sell you items. To continue, go through the door right next to Aria and to the next area, the cemetery.
You're now free to explore Akademeia, right after Moglin tells you how things work at the place. Basically, you're given free time in between missions, which will vary and is shown on the bottom right of the screen. You can ignore this and go straight to Central Command to get your next mission, but you can talk to people and do other things, like go to the overworld or do requests.
Each day consists of 12 hours and anything or anyone with a "!" over their head initiates an event that takes up two hours of that free time. Sometimes, these "!" will initiate a cutscene, but others will ask if you want to keep talking. The former are Sub-Events, while the latter are Talk Events. Go back to the main area and Aki will inform you that you can use the teleporter in the middle of the floor. You can find the various members of Class Zero spread around. Explore every part of Akademeia and see who you can help and what information you can learn.
So, feel free to do whatever you want with your free time, which is 6 hours for now. Once it reaches zero, then you will be forced into the next story mission. Head to Classroom Zero and talk to Kurasame, then use the front gate to go to the overworld.
Note: You're free to use whoever you want in your party, but only characters that participate in battle will earn EXP. Bringing along characters in your reserve allows you to switch to them at a save point.
Special Orders: In the Sorcery room, talk to the man with the Info icon above his head. SOs are challenges given in every area during a Mission. A little envelope icon will appear at the top of the screen and you can then view the challenge by pressing the touchpad (PS4) or View (Xbox One). The challenge must be completed while you remain in the area it was given. There will be a time limit or risk death but the rewards offered are well worth it. You also get a handy buff to help you complete the SO.