Loire is a great visit with kids, especially if you stay in a VRBO in Amboise itself. As noted, Da Vicini’s place (Clos Luce) is highly interactive, with large gardens and programs, depending on time of year. The town itself is great and is a good hub to bounce out to the other chateaus (Chenonceau, Chambord, etc.) If driving from Paris, stop in Chartres to visit the cathedral (and read the WSJ article on the US solider that saved it). I think I wrote up some notes on it in my “Adventures in Paris” thread here → (Adventures in Paris)
Giverny is a good day trip, especially if you think the kids will do well in Monet’s gardens. Very pretty, not too bad of a drive/train ride and you can backstop it with a visit to one of Paris’s most overlooked museums - “Musee de l’Orangerie,” which has the rest of Monet’s panels there. It’s also in Tuileries, which has a section of mini, individual trampolines in it to keep the kids going for hours 
Other things to do with kids in Paris - see whatever seasonal activity is going on at the top of the Tour Montparnasse. Highest view in Paris (iirc), and my kids have done everything from jumped on trampolines to roller-blading up there. Jardin du Luxembourg is a beautiful visit, next to the Odeon stop (see: Maison du Whiskey and the Le Comptour du Relais restaurants) and has a private park with tons of stuff to climb, as well as a giant fountain with toy boats in it. If the weather is nice, grab a couple bottles of wine, cheese, bordier and charcuterie and let the kids play around.
Amsterdam is another really easy/fun trip out of Paris. Take Thalys out of Gare du Nord (no security, decent wifi and you can walk around) and it’s a 3:20 min trip to downtown AMS. You can easily fly home out of Schiphol, as well. Highlights in AMS go for days - Rijksmuseum, Anne Frank’s house, canal rides, cheese, waffles, etc. See Blake’s threads for restaurants/stuff to do, as well.
San Sebastian is great, but not sure I’d do it in conjunction with a Paris trip. Another destination that you might consider is Corsica. Flights are usually super-cheap out of Paris (might be subsidized, given it’s a French territory), the food/wine scene is fantastic (Italy meets France), you’ve got beaches all over the place and lots of small towns. We got a VRBO, rented a car and hung out for a week. Only downside is that my kids were sensitive to windy roads.