You will visit the game's main dungeon tonight. In the lobby, you can use the clock to save your game and the blue door leads to the Velvet Room; the green portal thing on the right is the Terminal Point, but it isn't functioning now. You automatically head to the Velvet Room at first and have a little talk with Igor, who mentions that his real purpose will be revealed shortly. Once you're done, head up the stairs and you'll be transported to the first floor.
Listen to Mitsuru's tutorials on battles and play around a little bit with all of the functions to get a feel. You can only explore the first floor for tonight and your objective is to get rid of all the Shadows (you'll only fight Cowardly Mayas, who are weak against fire). Make sure you pick up the contents of the briefcases as you explore the floor. You may get an extra function after a battle called Shuffle Time, with a possibility of obtaining Personas, weapons, more experience, money or a chance to heal your party. You'll fight about four battles before you are told to split up your party (press Square and then first option).
If a party member finds the Access Point, head back to the lobby and you'll be done with the exploration aspect for now. Your Social Link for the Fool Arcana levels up to 2. When you get back to the lobby, you have the option to revisit the Velvet Room. Igor will reveal to you his true purpose; he is there to make new Personas for you by fusing together the cards you can find inside of Tartarus via Shuffle Time. I would hold off on fusing any yet until May 10, but feel free to fuse if you got any new Personas from a Shuffle. Whenever you're done, exit the Velvet Room and then Tartarus by choosing the option of going back to your dorm.
