We just got back from 2.5 weeks in Italy. Thanks to RS and this forum, we were well-prepared and had a great time. Thanks to everyone for their hints. Wanted to share a couple of thoughts I had that I thought might be useful to others.
Venice
I'd recommend staying in the Dorsoduro district - close to Accademia, so very close to everything but less touristy feeling than other areas we saw and very charming.
For the RS tour of lagoon islands - he recommends going to Murano, then Burano. I would do it in reverse - especially if you go in the morning. Burano is best early or late, with fewer people (this was at the start of our trip, so jet lag had us wide-awake by 5 am). So, starting at Burano with better lighting and smaller crowds would be good. Murano glass stores don't start opening until 10, so there's no point in getting there early.
Amalfi Coast
I would highly recommend the Mondo Tours from the RS guidebook. We had a wonderful experience with them. Also found Sorrento to be a charming home base.
Rome
I can't say enough good things about Rome. We were there for 5 days and it wasn't long enough.
Thanks to the person here who recommended the virtual reality guide at the Baths of Caracalla. That was a great suggestion and a highlight for the kids (and adults). The Baths were delightfully uncrowded and the VR experience brought it to life.
The tips about pre-booking everything made visits into big sights a breeze.
If you happen to be traveling with a boy, get him a Roma jersey on Day 1 and have him wear it every day. Roma fans are so nice to kids wearing the jersey (a baker gave him free food, we got waived to the front of the turnstiles for his jersey and more).