I too have been to Italy and Israel, although the last time I was in Israel I did not visit any Roman remains, so my memory of those are quite fuzzy. Maybe you’ve already been to all the places I’ve been. Personally, I loved walking through Roma, or Verona, or Aosta, and randomly finding some excavation of some Roman site in the middle of the street. I also of course saw the famous stuff - the Forum, the Coliseum, the Palatine Hill, the Parthenon, and so on.
My big Ancient Rome day, of course, was when I went to Ostia Antica, which, like Pompeii, is a whole Roman city preserved, but much closer to Roma, and thus more accesible. It got burried in Tiber river mud. The city today is dotted with little posts which comments on the city and Ancient Roman life in Italian and English. There is also some nice flora and fauna around the city as well - especially cats - and it is literally a breath of fresh air after the pollution of Roma. They also have a museum which stores the best artwork excavated from the city - probably to guard them from robbers.
I also went out to see some Roman emperor’s (Hadrian’s?) villa - while the bedchamber was interesting, the rest of the place was a bore.
Also, if you’re into Byzantine, check out Ravenna. The Byzantine stuff there is wonderful ; everything else about the town is awful, but the Byzantine stuff makes it worthwhile. All the pseudo-Byzantine stuff (i.e. Venezia) is ugly turkey - the real Byzantine artwork is beautiful. There too you can walk down the street and find a random excavation of Byzantine stuff which is not mentioned on the map.
However if you go to Ravenna on a weekend, expect the hotels to be booked out (it’s a beach resort), if you’re a vegetarian, live on gelato, if you’re a vegan, go to the supermarket to get bread fruits and veggies because I guarantee it’s better than any overpriced vegan food the restaurants will feed you. And if you want to meet some cold Italians, go to Ravenna.
I could talk on and on about Italy …