First family holiday in the USA — tips? advice? recommendations?

You might consider taking a train? Amtrak have pretty good value ticket deals for non-nationals, and it could be a good way to see a few places en-route (Chicago, Colorado, SF, Montana, Seattle, San Diego). Downsides are (possible) cabin fever and the slow pace of travel (three days coast to coast). Upside is you arrive in the heart of most destinations, and you get a much better feel for the country as a whole. (Oh, and you don't get all that grief that's become standard in US airports).

Spending 15 days travelling with Amtrak myself this April; NYC - Seattle - Vancouver - Portland - San Diego.

I'd put another vote in for California for the kids - San Diego in particular has lots to entertain kids. Great beaches, outdoor activities of all varieties, the naval museum on an old aircraft carrier, Seaworld (pricey but supposedly worth it), the Zoo is fantastic, and a wonderful climate.

Another California vote. The state has such diversity. We love San Diego, the zoo is great, and there is so much else to do. Eating out can be cheap too, we found a great Thai (called Royal Thai in the Gaslamp district) where dinner for 2 was something like $40 (2 courses plus drinks).

If you're doing theme parks I would suggest Six Flags and Disneyland but skip Universal. Universal has gone very twenty somethings lately, with little for the younger ones, or less young ones who're not into scary stuff.
I wouldn't do more than a day or 2 in LA itself.

Santa barbara is delightful, a drive around wine country is highly recommended.

Laguna beach is fun for a couple of days too. And then you have San Fran.......

If you're looking for internal flights I can't recommend Jet Blue highly enough. Budget prices, allocated seating, 38 channel seatback tvs and free soft drinks and snacks.