After the scene's you'll be at the Magic Academy Peristylim Suzaku - Fountain Plaza. Speak with the NPC's to gain some background information, then proceed to the fountain to meet Aki. You cannot access any other areas at this time, so head to inside the academy and go through the large doors on the left after you've spoken to all NPC's for background information. In Class 0's Lecture Hall, watch the cutscene, then speak with Moglin, who tells you to visit the cemetary first. Afterwards you can speak with the clerk in the middle of the room; there's not much to buy (or sell for that matter) though, however you also get a tutorial on how to set items in order to quickly use them in battle. Any consumable item can be setup this way; simply enter the menu, go to items and use the Set option to decide which item should be usable in battle for which character.
Note: All NPC's (in the central hall) have new dialogue after the above scenes.
Cemetary Scene
Enter through the door on the initial right to access the Garden, which leads to the Cemetary. If you are still using Ace you cannot access it right now. Head back to the antechamber with the Relic to get some more info regarding relics in general. More importantly, you can switch to Machina and go to the Cemetary to trigger a cutscene. This costs 2 hours of free time and grants you an X-POTION and an ELIXIR.
Crystarium
This area is accessed via the large doors in the central hall. You can find the Crimson Codex here (which is also accessible from the Main Menu), a book that contains an incredible amount of lore on the world of Orience. It helps to read at least a couple of entries every now and then, to catch the gist of the general background story. Also talk to all NPC's here and inspect the notice board on the wall to your initial left.
Military Command: 2nd Division
This area can be reached by going upstairs from the central hall and through the doors (above the blue statue). You can learn several interesting things about missions here, and you have the option to start a mission even when you still have free time left. Don't select this option right now though. It is also possible to replay missions you've already completed by selecting the Mission option from the Main Menu and then loading the appropriate save game. This is an excellent way to train your characters.
Magic Bureau
You can access the Magic Bureau from The Great Sigil. Here you can learn about S.O.'s, optional objectives that are given out to you during missions. Completing them grants you additional rewards, but failing them leads to almost certain death (you can try dodging the death spell that is cast upon you if you fail three times in total, but this is hard). You can participate now and still decline specific S.O.'s later. Speaking with Arecia takes up two hours of free time and grants you a HI-POTION, plus some info on why Class 0 isn't affected by Crystal Jammers. Yup, that took her two hours to say!
Arms Research Lab #6
You can purchase weapons and armor here.
Student Lounge/Salon
You can talk to NPC's here for background information. As with the other areas, there are also a few Mogs to be found here.
Terrace
You can speak with NPC's (and playable characters for that matter) here. Speaking to Machina triggers a short scene that takes two hours, so wait with doing so until you're completely done exploring the academy. You get a HI-POTION after the scene. (The NPC's on the terrace have new dialogue when your free time is up and the next mission is about the start.)
Chocobo Ranch
Speak with Obane to receive 10 MALE CHOCOBO'S, 10 FEMALE CHOCOBO'S and 3 GYSAHL GREENS. You can also do a test ride on a Chocobo at the field in the back. This doesn't cost free time.
Next Mission
When you're out of free time, head to Class 0's Lecture Hall to start the next mission.