I would stay with the Cullens, at least for now. (I'd have to go find a mate and travel after some time)
I don't know what I would do for my surname. Part of me wouldn't want to change it because it's a part of who I am. I can certainly understand why Rose didn't change her name. I think that, ultimately, I would accept the name Cullen. (though the family dynamic would be really weird because I'm madly in love with Carlisle)
Going along with that, it would be Carlisle who turned me, no question. (I think the question is whether I would be able to last eternity looking at Carlisle with Esme...)
Actually, Esme would be imprinted on by Embry (yes I know that makes no sense) and Carlisle would be with me! Yay Carlisle!