Persona 3 is a very interesting game, with many different elements that affect how the game plays. The moon, your friendships with people and even playing arcade games can have an affect on your stats in game!
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The school day
You play a school student, who you name. He has recently moved to the area and he's been put in a very empty dorm- the explanation for this comes later. Every day apart from Sunday (Japan, ladies and gentlemen!) you go to school and every so often your teacher will ask questions- some are just a case of paying attention whilst others are a case of guessing (or, in this day and age, google will suffice). You gain academics, social status and charm in school. You can also join clubs such as the Kendo club, allowing you to make new Personas (again, the game explains this in more depth).
At the beginning of the game, you have very few options with regards to what you can do after you've finished school for the day, but the options are expanded after you get past what is basically the "tutorial" stage. You can go to clubs, hang out with friends, go shopping and more.
In the dorm
Because there's only so much you can do after school, you might want to just go to the dorms- either to sleep, study or go dungeoneering. If you study, you'll stay up until you get too tired to study any more and then you'll go to bed. Sleeping can be useful, especially if you've gone dungeoneering the previous evening. Dungeoneering in itself is incredibly interesting and it's definitely one of my favourite parts of the game.
Without giving too much away, the dungeoneering in Persona 3 is similar to that of the Final Fantasy series in terms of the battle strategy. The enemies all have strengths and weaknesses which you can discover within the battle itself.
I quite enjoy Persona 3 and will definitely be trying it out on Hard mode (I played on Normal because I've played RPGs before).