If you're talking about university level study, I would suggest classics. Are you thinking in terms of majors? Ultimately, a lot of schools don't have a strong ancient history focus, it's one of the frustrating aspects of history departments.

Two things to consider, however:

1. How confident are you about your ability to acquire languages quickly and efficiently. At some point you'll need to take Greek and Latin concurrently, and speaking from experience, it ain't easy. Especially because you move through 2-4 grammatical concepts every week with attendant vocabulary. To put this in perspective, if you've taken Spanish, you know that the subjunctive is very important and my high school teacher spent a while teaching us about it and its various tenses. In Latin, the subjunctive is not only more complex, but you need probably burn through it.

2. Are you sure you only like Ancient Greece and Rome/mythology, or is that the era of history with which you are most familiar. Greece and Rome is what many historians and general lovers of history begin with, including me and many professors I have had. It's fascinating, and more importantly, there is lots of easily accessible information. You might find that you like something else!