I'm incredibly jealous of you! I was lucky enough to be able to spend half a year there in 2022. Mostly in Abu Dhabi but i got to spend a little time in Dubai for fun. The following is far too long but I wanted to get as much down as possible because I'm starting to have several friends ask the same questions.
Dubai
I agree with everything said above.
Small caveat for the desert excursions, and this is just me, but you get what you pay for. I have done several, but the last time I went cheap with a coworker and ended up in a Tahoe with seven people, very little ac and that drive out to the dunes can be incredibly tough. The car is constantly up and down and feels like it's always tipping over and can last for at least an hour. First time I have gotten sick in years so if you get car sick just something to consider.
The malls are incredible. The water show in front of the Khalifa at dusk is very fun to watch. There are lots of restaurants across from the mall that you can sit on the patio and get a great view. I think the last time I just went to the time out market, which is a normal food hall but has a perfect terrace for viewing without spending a lot of money. Other people probably know better but I also remember private boats going out to watch. Similar to taking a little boat with your friends to the bellagio fountain.
The Burj Al Arab as mentioned is incredible but I don't believe you can just walk in. I would always just get a reservation at the sky bar or gilt bar for drinks and appetizers and then they would let us in. I'm sure there are also several restaurants you could also make a reservation at to get you on the premises.
Ski Dubai in the mall of the emirates is very touristy but it was actually really fun to do once. Plus, after being out in the desert it felt great to be in 30 degree weather. When I went it was straight from the last desert ride and I just sat there for an hour at the top cooling off.
If you can't make it all the way out to Atlantis, "the view at the palm" is a great rooftop observatory that gives you good looks of the entire area. There are great bars and restaurants around there as well.
A little further south is the marina which is where I always stayed because I felt it had the most to offer. So many bars, restaurants, attractions and just street traffic.
Now to Abu Dhabi, which is where I spent most of my time.
With work i was staying close to the etihad towers which is as far west as you can get. The towers themselves are nice (if you saw fast and furious the towers they jumped from building to building) with the hotel Conrad having public access. There is a great observation deck at the top, along with a really good restaurant and bar on maybe the 60th floor.
The biggest thing to see on that far side is the presidential palace (Qasr Al Watan). More importantly, when the sun goes down they perform a giant light show outside over the front entrance for free. You can easily YouTube it to see if it's something you think is worth seeing. Just show up to the palace about an hour before they close, take the normal tour and then follow the crowd to the viewing area.
Right next to it is emirates palace. This is the famous five star hotel (which oddly was also in fast and furious). It's a stunning hotel that is absolutely worth the visit. It's on the water so if you get a chance to make reservations to any of the outdoor restaurants I would.
Staying in that general area the corniche beach area has several small water parks and beaches where the kids can play. Two great restaurants are the meat house and vertigo.
Staying on the main island and going further southeast you will get to the grand mosque. A must see. I think I enjoyed being there in the evening for pictures and with the dress code it's just more enjoyable in long pants. Right across from the mosque is the national aquarium if that is something your family wants to see.
Moving northwest there is Al Maryah island. This is more centralized and probably where I would stay to be able to get anywhere quick. It's where the giant mall is as well as the two great hotels. The rosewood has great live music.
Going more north east is saadiyat island, which is where the uae louvre museum is. I enjoyed it and it had a ton of great Egyptian history. It's fairly small and you can easily get through it in a couple of hours. Then if you want more beach time there are the st Regis hotel and the Jumeirah hotel right next to it to relax.
Finally going a little further you get to the famous Yas island. This is where Ferrari World and the f1 track are. Ferrari world wasn't really my thing but with kids I could see it being very fun. Lots of rides and go carts and simulations. You can go slightly south to the beautiful w hotel to see the f1 track. The biggest thing on the island is the newer Warner Brothers theme park which opened up maybe four years ago. I wasn't able to go but tons of people I met with kids absolutely loved it. If you golf the course there is very nice as well. To the far south of yas island is the etihad arena which usually has some pretty great shows so that may be something to look out for. When I was there they had everything from cirque du soleil, Broadway shows and I think Ed Sheeran so it could have a good concert while you're there. There is also the marina with several great restaurants to visit directly to the south of it.
I obviously have a lot more detailed recs so let me know if you need anything else!
