If you haven't already, check out: http://www.dubaitourism.ae/www/main/content.asp

If you are flying Emirates, they do arrange for the "desert safari" experience. Its really neat, you get a driver and a 4X4 - they take you out to the middle of the desert, set up a tent and a bonfire, serve Arabic coffee in traditional cups etc. - the whole experience, so to speak.

About ten years ago, DNATTA was their "preferred" travel agency, and they arranged this for about $75-250 pp depending on length of the trip. However, I know things have changed since then - but Emirates should be able to help you with this if its something you want to do.

Also, you can actually rent cars with drivers in most rent-a-cars in the UAE. Rates might be higher - about $50/day.

Both Hatta and Fujeirah (on the coast) are day trips worth doing - but you may not have enough time.

Make sure you check out local Arabic food - food is cheap and really good. Dubai is a really neat city - have a fantastic trip!