I've been hearing about these games for a long time, and I'm interested in checking on out. Do I have to play Persona 3 to play Persona 4? Or should I start all the way from the beginning?

Just watch the endurance runs, let Giant Bomb play the game for you.

Although Persona 4 contains a number of references to Persona 3, understanding of those references is not necessary to play Persona 4.
With that said, I'd still recommend playing Persona 3 FES before playing Persona 4. Not only will it make the events of September more enjoyable, but Persona 4 tweaks the gameplay in certain ways that may make it hard to go back to P3 after P4.

Well I like Persona 3 more than 4, but you aren't missing out if you jump to 4.

Well there is the PErsona series, the devil summoner series, and the Digital Devil Saga series all sort of part of the "MegaTen" games.  And of course Nocturne.
Persona 3/4 are more "action RPG + sims stuff"
DDS1/2 are more traditional JRPG turn based w/ cyberpunk+demon theme
Nocturne is like DDS+ Pokemon

Persona isn't action-RPG at all. It's entirely turn-based. Devil Summoner is action-RPG.

If you wanna get into the MegaTen metaseries as a whole and not just the Persona series, I would recommend you start with SMT3: Nocturne instead. It's the only game in the main SMT series (the direct successor to the original Digital Devil Story: Megami Tensei series which was based on a book and started the whole thing) that got localized, and it serves as an excellent introduction to what the series is all about. Stuff like Digital Devil Saga, Devil Summoner and Persona are all just spin-offs and didn't have SMT in the title when they were originally released in Japan.

So would it be safe to assume SMT games similar to America's Tom Clancy series? Some games being set in the same universe and others having similar traits.

Yeah, I guess you could compare it to that, but MegaTen is barely any different from other long-running RPG franchises like Final Fantasy, Dragon Quest and Tales in how its games relate to eachother. They often share common elements, have multiple established universes, and vary in gameplay as well.

I've played Nocturne, P3 and P4 and every game was separate enough to be stand alone, so I wouldn't worry about playing the prior game to understand the newer game. they don't follow one storyline.

the disclaimer is that I don't own devil summoner, just digital devil saga 1 and 2, nocturne, and persona 3 and 4.

You dont have to play any of the SMT games to enjoy them.  I'd suggest playing Persona 4 over 3 though because of the battle mechanic changes but dont neglect 3, its an amazing game in its own right.

"Persona isn't action-RPG at all. It's entirely turn-based. Devil Summoner is action-RPG.
"

"If you wanna get into the MegaTen metaseries as a whole and not just the Persona series, I would recommend you start with SMT3: Nocturne instead. It's the only game in the main SMT series (the direct successor to the original Digital Devil Story: Megami Tensei series which was based on a book and started the whole thing) that got localized, and it serves as an excellent introduction to what the series is all about. Stuff like Digital Devil Saga, Devil Summoner and Persona are all just spin-offs and didn't have SMT in the title when they were originally released in Japan.
"

I didn't play Persona 3 FES when I started Persona 4 and I don't see that it is that necessary. Persona 4 is the best of the two, though both are definitely worth owning. The only problem is that it is hard to find the third one. I got mine off of Amazon for a great price ($25) but I don't know if you could find it in store. Maybe used.