I recommend this Sea Recreational in Dubai and considered to be the one and only Dolphin Show in UAE, Dubai Dolphinarium. It's very affordable for an unfrgettable experience. This is great fun family event show and perfect with Kids and young at hearts. You will surely love the interaction with the Seals especially with the Dolphins.
So many Burj Kalifa, Dubai Mall, Miracle gardens, Ferrari world and yas Island in Abu Dhabi, Dolphinarium, Wild Wadi water park, Mall of the emirates, abras, dessert safari and many more. You can search the web to know more about these places.
A lot of good places you can visit. Should definitely spend some time shopping in Dubai mall and visit Ferrari museum as well. I would recommend to start your journey from Emirates airline and their amazing 1st or business class experience. You will never see something like that in any other airline. Should I mention all advantages of 1st class seat? They even have a freaking shower cabin on the board for 1st class travelers. It might sound expensive, but don't forget to check some awesome deals here, this website basically provides a possibility to buy 1st and business class tickets for best price!