Let's get Started!
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Fuse, Fuse, FUSE! It will really help you out in the later stages of the game.
Get your social link for The Chariot (Join the Sports team) to the max level so you will be able to fuse Thor. He was my go-to damage dealer for the final half of the game and during the final boss.
Speaking of Social-Links, get as many as you can! You get social-links by what else, forming relationships with others.
I would suggest getting all of your personality traits to their heighest level (Studying, doing karaoke at night, going to the cafe, and watching movies at the theater can help each trait) because they can open up social-link trees.
Grind those levels. It can get tedious, but it pays off towards the end.
Go to the bathroom before you go to Tartarus. It can improve your condition to 'great' every so often. (So does sleeping in class)
Make sure you always have a healer in the party to keep your HP looking decent.
You should stick to a main party your entire playthrough to make sure they level up and get all of their powers.
(I stuck with Yukari, Ken, and Koromaru. Ken and Koromaru won't be revealed until later in the game.)
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I hope this helps!
Now, If you would be so kind as to go ahead and buy Persona 4 right now.
Don't question it, just do it.