Looking to get into the Prince of Persia series - where to start?

The Sands of Time trilogy is connected, and The Forgotten Sands has been placed into those three as well.

I don't believe the other game is connected.

play the sands of time, than 2 thrones, then warrior within. Stop. U can try PoP2008, the story is different one, but i liked it. The forgotten sands is a piece of crap.

It's divided in:

Original trilogy
Sands of Time tetralogy
2008 game

These miniseries are pretty much separate from each other. If you're feeling in retro mood, pick up the first one, and maybe you'll like the second one too. (PoP 3D sucks though)
For Sands of Time, just play in order of release. I never played Forgotten Sands, so no idea if it's any good. I almost called it a trilogy because I forgot about the last one.

For the original trilogy, I've only played a part of the original Prince of Persia (1989), and none of the two that followed (1993, 1999) and also an old mobile version of the game (pre-smart phone era). If you like oldschool games though, they might be worth seeking out.
Otherwise, start with Sands of Time (2003 and continue through the rest of that trilogy (Warrior Within, Two Thrones)). Warrior Within particularly seems to get detractors because the Prince gets all aggro, but I liked it.
PoP 2008 has very little to do with those games other than actual gameplay, so if you want to play a more modern engine, you can play that one whenever.
I haven't played Forgotten Sands, the most recent one, at all yet.

You can play them in order though forgotten sands is in fact next to sands of time they don't really connect. POP of 2008 you can play anytime cause it has nothing to do with the others.

Definitely play the trilogy in order, the story of those 3 games connects together so well if played in order. If you do not play it in order you will be confused and it will break the story completely xD

Sands of Time, Warrior Within, Two Thrones, and Forgotten Sands are all connected and meant to be played in order. The other Prince of Persia game play somewhat similarly but isn't connected story-wise.

Play the games connected to the Sands of Time. Definitely the best games of the series in my opinion and you look more and more badass as you progress through the series!

Start with the Sands of time trilogy, in chronological (heh) order. Skip the classics, because they didn't age too well, and because without nostalgia goggles they're only decent and no more, and they also don't contribute a whole lot to the Prince's background. You should also skip PoP 3D because it's pretty much a badly crafted raft made out of giant dried pieces of turd, floating on the piss ocean of early Tomb Raider ripoffs. Seriously, it blows.

Only new trilogy + PoP 2008 one. Those are ones I enjoyed most (2008 is surprisingly good if you are into darker stories).
Forgotten Sands is terrible, I managed to finish it in 5-6 hours and story isn't really related to other games. Also it was boring and predictable as hell. Don't touch 3D one, it's worst PoP game...

The Sands of Time, Warrior Within, The Two Thrones.

Only these ones :) (They're the best in my opinion)

P.S.: I'm not counting the 2D ones :)