What is dungeon crawling? I'm not used to RPG's in general besides for Pokemon & Paper Mario

It's a very awesome game that you should be playing.

I would highly recommend you to check that game if you like rpgs ofc.The gameplay is one of the best when it comes to jrpgs, I just don't like the characters too much they look too normal...maybe because they're normal kids to begin with but that might just be me lol.

It does have some simulation elements in it. Specifically, social simulation in a sense that you can make and improve relationships between characters and such just like how you would interact with a person in real life and participate in various social activities with them ... 

Which game is coming to the PS4? P5?

How do I play the other games? (I'd assume I should start with the original)

A dungeon crawler is a type of scenario in a fantasy RPG in which the protagonists navigate a labyrinth like environment for the purpose of battling various monsters and looting treasure ... 

Persona is way too cool. Great RPG franchise!

The year 2000 called. It wants its "not instantly looking at Wikipedia" back.

What the poster just described to you is Persona 4: The Golden which is only available on the vita which is an enhanced port with a moderate amount of new content but if you have a PS2 (which is unlikely judging from your account on platform ownership history) then you can play the original Persona 4 along with Persona 3 and it's enhanced rerelease Persona 3: FES. 

Persona 5 is indeed coming to both the PS3 and PS4. 

You don't need to play the other games to understand their respective stories in each game. It will help on understanding the background but it isn't necessary just like Final Fantasy ...

Start with 3 or 4 (I think 4 is better and a must play) but the stories don't really tie together so you could play either. Persona 1 and 2 are old and outdated games now.