The major gaming site Operation Rainfall had time with Persona 5 and took away a lot from the game. Some impressions on the games demo dungeon, some plot details and and a few mechanics are detailed in the article. It is a comprehensive breakdown of its E3 showing and I strongly suggest giving their article a read if you want to learn more about Persona 5, which will be at the source link below.
Source: Operation Rainfall

The scene opens with footage of the cat Morgana hanging out in the protagonist’s bag. This is where she will usually be as she comments on the various activities you participate in. She is usually hiding out of sight, since pets are not allowed at school.
The protagonist has a certain number of days to complete a mission before he is expelled. This countdown can be found in the top right corner of the screen and will be shown in red the closer it gets to zero. This wouldn’t be such a problem if there weren’t so many distractions around town. The player will be able to do lots of activities, including going to the batting cage (where ATLUS humbly wrote their name on the bat), attend a part-time job, or even go to the movies. Additionally, there aren’t any gun retailers in Japan, but the player will be able to purchase air soft guns as weapons. These activities, if done currently, give you the points to upgrade your skills. There are five skills: Knowledge, Guts, Charm, Kindness and Proficiency.
The dungeon we see explored is the Kamoshida Palace. The halls are crowded with shadows that the player will be able to sneak up on to cause starting damage to the enemies. Sneaking is fast and allows the player to hide behind desks and corners with the press of a button. Outside the palace are the twins from the Velvet Room, Caroline and Justine. We were not shown how big of a role they play, but it could be a way to swap out party members before entering the dungeon.
Players will come to puzzles in the dungeons that need to be solved to progress. You are given the option to solve them on your own or use a “Third Eye” that shows you clues, players can use this at their leisure with no real downside. Additionally, map pieces can be found in various spots that (we assume) if every piece is collected show the entirety of the dungeon.