Seriously, when I am in Italy, I think of ancient Rome most of the time. The country is just chock-full of Roman structures and you won't be walking for a long time before bumping into one.
When holidaying in Italy I had the luck to pick the day trip one day. Really glad I found Paestum (an ancient Greek city): it is every bit as captivating as Pompeii imo.
Went to Italy for the first time a few years ago and picked paestum randomly when we needed a break from Naples. Went back last year and will probably go again.