Fine Tuning Family's first Italy Trip this summer

Lucca, yes to riding bikes up on the city walls which have a park covering them.
Yes to a walking tour in Florence. Get tickets ahead for L’Accademia to see Michelangelo’s ”David.” Our grandkids loved seeing David and it doesn't take very long once inside the museum. Kids might enjoy Galileo Museum.
Also check out Leonardo da Vinci Museum for the kids.
Tuscany- the kids loved it all especially Florence, Siena, Volterra, Panzano.

I'd flesh out the Naples portion a bit--it's such an overwhelming place but with so much to offer. I would not necessarily want to spend the whole time there doing day trips. Archaeological Museum? Underground? Carvaggio? Castles, Forts. There is even an archaeological site you can snorkel over!
https://www.napoliunplugged.com/
http://www.parcoarcheologicosommersodibaia.it/parco.php?id_lingua=en

It was July when we were in Naples and very hot. We did the underground tour because it was about 20 degrees cooler underground. But we really thought it was interesting and I would think it would be something kids would like as well. We did it during the heat of the afternoon and afterwards it was much cooler and we just wandered around.

When in Naples, you might want to consider Herculaneum in addition to Pompeii. We did one in the morning and the other in the afternoon. Two totally different sites that were ruined by the same event. Google it and see what you think.

The best stuff from Pompeii is in the Archeological Museum in Naples. The Pompeii ruins are impressive and a great place to visit, but Ercolano/Herculaneum is better for me because it is smaller, and really is in a hole in the volcanic ash which gives a perspective on how much came down and covered the town. The boat sheds with their skeletons are compelling. The Vesuvius trip is cool..... The last eruption was only in 1944 so it is an active volcano.

All great places, but why are you skipping Rome, it is wonderful.

While in the Naples area, definately do Capri and Sorrento. The Amalfi Coast if you have time. The beaches in the area are not the greatest.

Lucca and Siena are great and half a day in Pisa.

We did a nice bike tour in Florence, if you like cycling.

Thanks for all the feedback so far! For those who've wondered what went into my choices, I have family in Naples area that I am long overdue visiting, ans want to explore a bit there. My family is more used to beach vacations and are not really big on visiting cities, which is one of the reasons we're skipping Rome--plus I've been there several times, and with the 2 nights I had left for the vacation, I thought Venice would be a more manageable choice. They are also not super excited about museums, but I do want them exposed to the incredible history and culture here. So maybe Herculaneum instead of Pompeii? Although I'm interested in the thermal baths of Ischia, is Capri better suited for a day trip? I'm thinking of taking the train into Florence rather than drive for our day there, and like the idea of a 3 hour walking tour.
I appreciate all your comments and suggestions.

Ischia will have better beaches for the kiddos--I'd do that instead of Capri unless you were interested in hiking on Capri.
I think for a first trip (for the fam), I'd go for Pompeii--there is just nothing like it in the world. I can't imagine kids not loving it. Though certainly if time is limited or it is just too hot for Pompeii, the smaller more intact Herculaneum is a good option.
Another outdoor option is Phlegraean Fields.