Airline...
If you're not the type that can sleep, get an aisle seat and move around a lot. Otherwise, what everyone else said. I'm guessing you'll be going on Emirates. They're a great airline. Service, even in coach is good. So is the food. They'll leave some sandwiches out that you can grab as you want between meals. Large selection of movies to watch.
Dubai itself...
As AOA touched on, be careful of your western ways. Don't be drunk in public. No cussing in public. Ensure your wife is covered up. No bare legs, except at the beach possibly. No public acts of affection. Not even a hug. Holding hands is tolerated. Kissing is not.
Cabs are relatively inexpensive and are heavily regulated. If you hop in a cab, you can be ensure of a safe and honest ride. At the airport, you may be ushered to a non-descript black car that is actually a cab. Despite what they tell you, they are more expensive than the Dubai cabs. If it makes a difference to you (about $10), insist on the Dubai cab. Once you're in town, you can pretty much pick up a cab within minutes (if not seconds). They're everywhere. However, the big malls and major hotels (including the Hilton at The Mall of the Emirates) have rules that the cabbies are supposed to follow as to where you can be picked up. The cabbies can get into serious trouble (big fines) if they break those rules. So, just be aware of that.
Mall of the Emirates has the indoor ski slope. If you're interested, go for it. You'll have to provide your own hat and gloves, but everything else is provided.
The Dubai Mall has a rather large aquarium. If you're interested you can scuba dive in there. But they won't let you take pictures unless you take there photography class.
Something no one has mentioned yet is the hot air balloon ride. Since you'll be there in December, they'll probably be doing them. Just like the US, they tend to launch at dawn.
Check with the concierge at your hotel to see what events are happening. Lot's of concerts and sporting events going on all the time.
EDIT: Forgot to mention. LOTS of great restaurants in Dubai. You'll recognize a lot of the chains you're familiar with, but don't be afraid to experiment with the local restaurants.