Must dos i guess depend on your interests and time of year but here's a few of mine:

DUBAI MUSEUM, BASTAKIA AREA AND CREEK
The museum is highly recommended, it's really amazing to see how the city's developed over the last 50 years and find out a bit about desert life.

Take a walk from Dubai Museum, through Bur Dubai Souk to the creek. Along the waterfront you'll find numerous cafes and if you're still curious about more historic Dubai you can visit Al Maktoom House and The Heritage/Diving Villages.

From the creek it's possible to take an Abra (extremely cheap water taxi) over to Deira where you can visit the Gold and Spice Souks.

WILD WADI WATERPARK
This is a fantastic waterpark with many different rides/attractions including some that actually jet you uphill. They have a well organised system where you charge a wristband for spends and they refund any credit when you leave. The only downside is the food - it's the usual, burgers, pizza, etc that these type of places offer but with the great restaurants in Dubai it seems worse than ever.

GREAT RESTAURANTS

JAMBASE at Madinat Jumeirah does great food, especially the american style fish dishes and excellent live jazz. When visiting any of the restaurants at Madinat Jumeirah we always stop at Bhari Bar (Mina A Salam Hotel) for a drink on the terrace - great views of The Burj Al Arab over the water. Bar Zar is a livelier bar in this complex which is a good pre club venue for Trilogy.

MUMTAZ MAHAL at Arabian Courtyard Hotel is a fantastic indian restaurant, very popular with the local indian community. The best night to visit being Friday night when they have a resident indian band and dancers in traditional costume.

SPECTRUM ON ONE at The Fairmont Hotel is my favourite Dubai restaurant, the nearest desription i can think of is a very upmarket food court. They have 6 open kitchens offering menus from around the globe and the only problem i ever have here is deciding what to order from the vast menu! Cin Cin in the same building is a great little bar - can only be desribed as 'uber cool' with a great wine list.

BASTA ART CAFE near Dubai Museum is a lovely little courtyard cafe inside a traditional house. It's more of a brunch/lunch venue but they do delicious homemade food and a good range of smoothies, freshly squeezed juices, etc.

AL DAWAAR (the revolving restaurant) at Hyatt Regency Hotel. This is a buffet restaurant, something i'm generally not keen on but well worth doing for the experience alone. The restaurant has large glass windows and turns 360 degrees while you sit, dine and take in the fabulous views.

There's also great shopping in Burjuman, Mall of The Emirates and Wafi City, etc. Many enjoy the famous desert safaris but personally i found this a bit of a tourist trap and not something i'd recommend.

Hiring a 4x4 and driver to take you across the sand dunes in the desert is a must do - scary but thrilling experience.