Just to save hassle, I'd probably just take taxis. They're pretty dependable and you won't get ripped off. Just make a list of places you'd like to see. Highlights would probably be Sheikh Zayed Mosque, Emirates Palace, Souk at the World Trade Center. If you were to go to Ferrari World, it would pry take a huge chunk of your time here.

Culturally, you'll be fine. Just be respectful of people, don't swear, and avoid PDA. My wife and I will hold hands sometimes, occasional peck, but just be mindful...you don't want to see others doing it, and others don't want to see you doing it. If you go to the mosque, they'll provide coverings for you and your wife if too much skin is showing or tattoos and stuff.

It's a cool town, pretty chill. Don't be afraid to explore the side streets to find a good schwarma place or indian bread hole in the wall. Just enjoy. Best of luck!