I can't comment much on Bahrain, but I've lived in UAE most of my life. IMHO Burj Al Arab is overrated and pretty fucking expensive if you're taking your family (more rooms I'm guessing). Go to the Jumeira Beach Hotel right next to it. Almost the same experience, same beach and gives your wallet a break. Burj Al Arab's grandeur wears off after a few hours. You can pay the 150DHS and go check it out because it is phenomenal inside and there's a great restaurant in there. Rent a car and drive down to Abu Dhabi (2 hours away tops). There's a place along the shore called "Kaser al Amuaj". Amazing man made island. There's an old wooden ship on there that used to be used to get pearls. It's transformed into a great Arabic restaurant. Mostly Khaleeji food. The Marina mall is pretty cool too, it's just up the street from there.
Check out the new and old Corniche, great place too!
Same city, there's a restaurant called "Zahrat Lubnan", best lebanese food I've ever tasted.
If you want to spend a night in Abu Dhabi I really recommend the Gulf Hotel. Rooms aren't AMAZING, certainly less nice than Dubai hotels, but the pool and beach are priceless.
My home town is Al Ain (1.5 hours from Abu Dhabi or Dubai) and has one of the only mountains in the region. Jabal Hafeet is it's name. There's natural sulfur hot springs there and I would recommend you spend an evening there, it's amazing and very relaxing.
If you feel adventurous head over to Diba, it's on the other side of the country (keep in mind it's a SMALL country) and is a great way to enjoy the indian ocean.
In Dubai, check the Speedway there, great cars passing through there. I'm not going to go on and on about the city because it's probably one of the best documented cities in the gulf in terms of attractions. There's two things I'd recommend, the Irish Village for some nice bars, and if you're bringing kids, Wild Wadi (I haven't seen another water park in the world that was as much fun)
If you need more recommendations just let me know and I can certainly help you out.
One word of caution, watch the roads. A lot of the young kids in big SUVs seem to think road signs don't apply to them and will do double the speed limit, and watch out for night driving. There's even more of them.
The places I recommended aren't crazy expensive and give you a better experience of the country. I don't like going to the super expensive french cuisine places, because I can get that when I go to paris.
If you want my recommendation, I recommend you don't stay too long in Dubai because most of your time will be stuck in traffic. The extravagance wears off quickly so I recommend going to the other cities that offer a more genuine experience.
PS. If you're taking the wife there, jewellry is CHEAP!
Best, Walid.