My suggestion,,

Do maui or kauai. I spend around 6-8 weeks over there and have been around all the islands. Oahu traffic is one of the worst on the world. U will be carrying the kids everywhere,, can’t get anywhere and will never really C much of anything other than your hotel room and maybe the dirty beach…

Go to Maui and do the kanapalii area. better family area.

If you have ten days in which to see Hawaii,I’d seek to see if they have a tour boat and enjoy the trip and view from ground level.That may even be cheaper!?
I lived on the isle of Oahu for a couple of years,never got to go to any of the neighboring islands but what I seen of Waikiki and Honolulu, it was heaven.
MAHALO!
Mele Kaliki maka, me ka houli makahiki hou!
Ask what that means while your there.

I frequently travel to HI, and I found the cheapest way to book inner island trips is to book ahead of time through a travel agent. 3 islands might be a little much to see in 7-10 days with 2 kids!! Maui is our favorite! Lahaina (west coast) is beautiful!! We love the Royal Lahaina’s luau the best, all u can eat and drink and kids were free! Maui also has the maui ocean center that is awesome for kids!! Oahu was ok, very city like with lots of homeless people in certain areas. If you go to oahu, pearl harbor is a must see, very educational. We also swam with dolphins at the dolphin encounters that was my favorite!! There is a company called Polynisean Tours that pick up and drop off right at your hotel. They have a website and a ton of activities!! They are island wide I believe. They also have a sunrise tour that is just unreal!! b(take warm clothes for that cause it is on top of a dormant volcanoe) The road to Hana is awesome but take a tour bus if you are not a good driver!! The roads are very twisty and narrow and it is a 4 hour drive up and 4 hours down!!!! But the black sand beaches and 53 waterfalls you pass are gorgeous! As far as Kauai, We will be there in Jan. on the east side known as the coconut coast. The south side is known as the “sunny side shore” Tons of sunshine!! Things to try if you like drinks are Lavaflow and Blue Hawaii!! tThose are my favorite!! Find the ABC stores for things to take home!! They have gifts for cheaper than other places. Also Hilo Hatties is a cool store. If you like coffee Kona is the best!! I priced our vacations all different ways and the cheapest was through an actual travel agency. It was more expensive to fly inner island once we were there rather than book ahead for our package deal. Have Fun!! Aloha! (Goodluck on that 10 hr flight with the little ones!! We leave ours with mom and dad!!)

Flying between islands is not cheap. Try a cruise.