I believe that goes without saying.
I’d go with Rome, because I could not resist the chance to “invent” more advanced military hardware and see it put to good use. Eventually I would hate myself for creating engines of destruction, and then I would become a tortured poet and create classic works that would resonate down the ages.
Ditto!
There was a young lass from Pompeii
Who really was quite a good lay.
2 denarii a feel,
but to do what you will,
an aureus, spare as, and Olè!
I’ve got a partner in crime! Hell, sign me up right now! 
I love Latin love poetry.
I’m brushing up on my Ovid as we speak. You’ve got the whole war machine thingy figured out, right?
I tried that with Sappho and got slapped…
So did the Romans, anywhere from once to five times. In fact five times watered wine (1/5 wine, 4/5th water) was often served to children. Few adults drank wine that wasn’t watered down 3-1. True, they drank a lot of it.
Oh yeah. I’ve built models of trebuchets and onagers, and I know the principles behind ballistae. With a little knowledge of modern engineering, physics, and chemistry, I’m sure I could improve on things greatly, assuming I didn’t get myself accidentally blown up in my alchemist’s lab.
In between all that I’d see about building a Babbage-type math engine, but that might prove too difficult with the technology at hand.
I too, would jump at the chance to play “inventor” in Augustan Rome. It would be nice to have either a few months’ preparation to learn as much about modern engineering as possible, or failing that, a few dozen textbooks. I’d try to work out an arrangement that allows my “inventions” to benefit both myself and Augustus, just to make sure that he doesn’t mistake me for a threat.
Even though it has its drawbacks, especially for a woman, I’m going with Greece. I like the fashion and architecture, and frankly, I could do with a few years of lounging around the house. I’d spend my time reading – you know what, memorizing – some of the Greek texts that have survived only in fragments or not at all. Then, if I had to temporally relocated again, I’d write them all down and be totally the most amazing Greek scholar ever. And even if not, it would be cool to read them.
What types of wine now in existence would be closest to what the Romans drink? Chianti, Merlot, Bordeaux- any idea which would have a similar taste? Did they have both dry and sweet wines?
Rome(under Claudius), chariot racing,the games,theatres,libraries ,baths,public spectacles (triumphs,temple processians )being able to travel over most of Europe reasonably safely,no chance of nomadic invaders until much later and on top of that a free grain issue.
See here for some detail on those questions.