We went to Naples and then Rome last summer.
Lots of Gelato. Our dc (then 9,11,13) enjoyed eating out, trying different pasta etc. (a wee bit out of the centre makes eating much more affordable!).
We did Coliseum, Pantheon, Piazza Navona, Spanish Steps, Vatican (museum was definitely a mistake with dc, even dd1 the art lover!) quite a lot of just hanging out by fountains (good for a bit of shade!!), taking funny family photos, a bit more gelato, somehow it all seemed very relaxed.
I don't know if you'll have time as it's a bit far for a day trip, but we loved Pompeii, Vesuvius and other stuff in the Naples area, and we had a villa with a pool there :)
Have a great time, it was a fantastic trip for us.