You begin in the Velvet Room, where you're talking to Margaret, who mentions about something that's about to happen to you. You'll then get to name the protagonist of Persona 3, so choose that and then the game will begin. Watch the scenes, choosing whatever you like as it doesn't really have any impact.
Eventually, after introductions and such, all of your party will hear a bell and then meet up with Margaret at the culture festival. She'll bring you to the Velvet Room and the bell will be heard again. You head back to the school and everyone heads after Teddie and Yosuke to the Cosplay Cafe, which is when you'll be in control.
All you can do at the moment, though, is either go to the Cosplay Cafe or save your game, so go ahead and save for now. Head to the cafe and you'll find that it's been replaced by something else. Everyone heads inside and it turns out that Personas can be summoned here, along with Shadows being present, so the group decides to investigate some.
Your party consists of the P4 Protagonist, Chie, Teddie and Yukiko. Ignore everything else for now and head straight forward, where you'll find a door. Go through it and Yukiko will give you 5 Medicines, which will be useful should someone get hurt. Anyways, there's only one path for now, so take it and you'll get thrown into a battle.
You're forced into a battle with three Lying Hableries, which are weak against Bash, Fire and Lightning (you have two of those right now in your party). The enemies only have two attacks of their own, a physical one, which does little damage, and Bufu, which does a lot of damage. Using one of their weaknesses will put that character into a Boosted state, which has two advantages.
A character in Boost will act before everyone else on the next turn and all of their skills, both HP-costing ones and MP ones, will cost zero for that turn, unless they use another weakness. However, should the Boosted character get hit with an enemy attack before they can act again, then they lose their state and everything returns to normal.
After the battle, continue going through the labyrinth and into the next room, where you'll meet Zen and Rei, who will join your party. It's now time to leave, so go back the way you came, where you'll be forced into another battle, this time against two Lying Haberlies and a single Cowardly Maya. You already know the Hablerie's weakness, and the Maya is weak to all four elements and dark.
The Shadows will drop a Shadow Piece after the battle, which will become useful in a little bit. For now, exit the dungeon and return to Yasogami High. You'll eventually return to the Velvet Room and gain access to a shop and rest area. The shop will use the item drops from Shadows, although you'll have to pay money. The Nurse's Office will also cost money to heal all HP and SP based on your protagonist's level (100 per level, so at level 22, you would have to pay 2,200 to use the Nurse's Office).
Next, you'll learn about setting up your formation, using many factors, such as character strengths, like their ability to attack from the back row without mitigating damage, as well as HP and defense. Make your choice and feel free to visit both the shop and Nurse's Office to prepare for your next entry into the labyrinth.
Now, you should have gotten an item called Goho-M along with the Long Sword. The Goho-M will allow you to teleport back to the school no matter where you are in the labyrinth, so don't be afraid to use it and make sure to always keep one in your inventory.