I spent five enjoyable days in the UAE in 2003, most of my time was spent in Dubai.
Dubai is split into two parts by "The Creek", an inlet of the Gulf, and the easiest way across? An Abra - for a few pence you can jump on a small boat with twenty other people, and in five minutes you'll be on the other side.
I stayed in the Alkhyam Hotel, a small hotel just outside the Gold Souk, and if you are into gold - then that's the place to go looking for it - anything made out of gold is sold there!
The spice Souk is also a great place to wander around especially in the evening - you cannot imagine the number of smells that you'll experience!
A good way of getting your bearings in the city is to take the open top Big Bus tour, it covers both sides of Dubai Creek, and the bus ticket gives you free entrance into the Dubai Museum, and the Sheikh's House.
In Dubai there is an ancient form of air conditioning, because of the breezes from the Gulf, the locals added a tower to their houses, with open windows, and a 'X' shaped cross section which directs the breeze downwards, into the house.
Wafi City is a huge shopping complex which is a stopping point of the Big Bus, and it worth a look, even if it's just to escape into air conditioned luxury! I've never seem so many "designer" shops under one roof!