We're visiting Rome. What other cities in Italy near Rome can we visit at the same time?

By bennyzero, you can't be in the same place at the same time. It's one of those physics thingys.
There is Naples, Capri, Pompeii, Gaeta, Siena and Assissi just for starters
Florence is a very easy train ride from Rome, and a MUST see. And after you have checked out Michaelangelo's David, don't miss going to Vivoli for gelato, they have the best on the planet.
I would head south (roughly 2hrs) and check out Napoli, Capri (absolutely beautiful).....here you could take a boat and check out the grotto.......you'll need to get into smaller boats and row on through The Amalfi coastline is stunning. Pompeii......let's not forget the volcano that caused the destruction Mt. Vesuvius. Also, just before Napoli is Caserta. In Caserta, is the Royal Palace. You could spend the day there alone. Everytime we go, we pack a lunch and have a picnic at the Royal Palace. It's an amazing tour inside and out. Sorrento is not far...it overlooks the Naples Bay. It is a hot spot for vacationers. Good luck....my family is from the area I described. I promise that you will have a good time. Just be careful in the city of Napoli.....gorgeous at night, but like any other city....there is crime to be found. Careful here as you those "gypsy" rumors are well known to be true. Travel with someone in the city, who knows the city well.