At the suggestion of another FTer, I attended the Sheikh Mohammed Centre for Cultural Understanding. They offer a tour and a traditional meal for breakfast, lunch, or dinner. They give a talk about Dubai traditions and answer any and all questions about Arab culture and Islam. This meal, tour and presentation was the highlight of my trip to Dubai. You will leave with a cultural understanding that the average tourist never gets. They have a website that should answer any questions you have about the program.
Edited to add: You can pick up an Entertainer book in Dubai which offers hundreds of discounts. It is rather expensive but using one coupon could pay for the book. There are many BOGOs. If there are two of you traveling it will definitely pay for itself. The SMCC, desert safaris and hundreds of restaurants are (or were) in the book.