I think the Klingons will be in the second film, but I'm iffy on whether they should be the main villains. I think it would be interesting if the Romulans were the go to, stock villains of the new movies like the Klingons were in the TOS movies.
I don't want to see Khan again, I just don't think they can recreate that magic or add something different to the character like Heath Ledger did with the Joker. Part of what made TWOK work was the history between Khan and Kirk. It was something palpable and I don't know if they can recreate that. I wouldn't mind seeing them introduce Carol Marcus though.
There are other TOS villains they could go with like Gary Mitchell, who would have a more personal tie to Kirk, or Garth of Izar, who Kirk admired. There's the M-5 computer, Trelane (a possible Q?), and others that could get more attention without retreading Khan. Nero to some extent was a Khan retread already so I would like to see them go in another direction.
Two TOS races never explored were the Tholians and Gorn. Both could be awesome adversaries. There's also the Orion Syndicate.
Personally I would like to see a remake of "The Enterprise Incident", but with a more beefed role and a name for the Romulan commander. Perhaps throw Section 31 into the mix and it could be a nice espionage story.
It would also be cool to see totally new villains, but there is a part of me that would like to see the Romulans, Gorn, and Tholians get more development, also to see Section 31 in the Kirk-era.