I'll join the crowd in suggesting you avoid Cinque Terre in early September. Since you're arriving in Milan, you could easily substitute lake time. Lake Maggiore & Lake Orta are beautiful, Orta will be the quieter of the two. Might be too sedate for little ones, since key activities are walking and eating. But Lake Orta has a great campground with cabins that has a nice beach on quiet, shallow waters and a playground. Could be fun. Or, you could head northeast to Lake Garda, which has a great water adventure park as well as beautiful towns, great food, all the Italian things you're probably looking for. Maybe visit Verona while at Lake Garda for a bit of (beautiful) city time.
For a second home base, absolutely go to Lucca. It is wonderful with and without kids, big enough to see things and have fun, without the big crowds to push through. I spent four nights there on a 2016 trip and loved it.
If you still want to go to the Italian Riviera instead of the lakes, look further west around Genoa. Take the kids to the biggest aquarium in Italy, then settle into a smaller Riviera town such as Santa Margherita Ligure.
These are just a couple of thoughts...there's much more but I think you would be wise to stay out of the big cities until the kids are older. Heat and crowds are no fun for littles (or bigs.)