With a new Twisted Metal game coming out next year for PS3 and me not playing any of the previous titles, which ones in the series should I play? Which ones are available on PSN?

My favorites were Twisted Metal 1 and Twisted Metal Black (PS2).  AFAIK only Twisted Metal 2 is available on PSN but I hear TM1 is coming soon.  Most people seem to prefer TM2 over TM1, but I felt TM2 was very janky feeling compared to 1.
 
I'd suggest you try Black as that's going to be more solid in most ways.  You'll just miss the amusing FMVs of TM1.

Play TM1, 2, and Black. There are no other games. I am going to say that again.
 
 
THERE ARE NO OTHER GAMES.

Just play Black, it'll closely mimic the new one.

If you want two games, make it 2 and Black, if you're sticking with just one game, go with Black. Head-On is actually pretty great too, AND it has the bonus of all the original Twisted Metal 1 live action endings that David Jaffe, by his own admission, took super-duper seriously at the time. They are HILARIOUS.

" Just play Black, it'll closely mimic the new one. "

Ideally, you want to see all of the original games, just to understand how necessary it is for this game to be a rousing success lest it fade away entirely.  
 
The best games in the series are 2, Black, and 1, in that order. But at this point the original and its sequel are sort of difficult to look at. Much less to play. You might be alright just playing Black and getting a sense of the series's humor, and a sense of its mythos. Then again, I feel like the only reason that stuff works so well in Black is because it's a total reinvention of the mythology we'd come to know so well over the course of the previous games.  
 
Tough question, dude. If you can snag copies of the original two games for cheap, totally go for it. They're really weird experiences, even today. But if your search runs dry there, Black on its lonesome will probably be enough. 

I'll take any excuse to reminisce about TM2, it was one hell of a split screen game. Also this: