We've been to Rome 3 times, Athens 2. We greatly prefer Rome. Just more to see and do. I don't want to seem negative about Athens though. You should see both, but IF you can only visit one-----
Include the Vatican, and hire a guide. The Sistine Chaple, Michaelangelo!!!, Colisseum, pantheon, Trevi Fountain, etc.
Florence is a day trip. 1 word, David!!!!! They don't make humans like Michaelangelo any more.
The Amalfi coast is worth at least two days. Pompei is unbelievable. Positano, and other towns are beautiful. The food might be the best in the world. Certainly, the pizza is.
And if you can get there Venice is a "must see". Several days in the area. Lake Como.
Greece is nice to, but you might want to consider a cruise that will visit several of the islands and possibly Ephesus. Even Istanbul.
Rhodes, Olympia, Delfi. You might get ruins overload?
The Mediterranean, a warning, They give masters and doctorates in pick pocketing. Especially in Rome.
But, much less violent than the US.
Don't go in June, July, August. Too hot and too crowded.
Enjoy your trip.