My training, in addition to Greek and Latin, required at least dictionary level familiarity with French, German, and Italian. I would stress the German - it's the most foreign to me, but actually kinda fun. I wish I had more time to spend on it.

Definitely study abroad if you can. You can improve your languages and travelling is always worth while - you meet interesting people and learn interesting thing. If you're thinking about the whole of antiquity, you might want to go to Rome, or look into Roman Britain, or visit Istanbul.

I haven't been to Greece for more than a decade. I have no clue what kinds of events etc. are going on now.