The ship docks in Port Rashid. It is too far to walk into town. There are courtesy buses which drop you at Burjaman shopping Centre. Taxis are extremely cheap though in Dubai. We went to the shopping centre and bought the "Big Bus" tickets where you can hop on and off as you like. there is a City bus and Beach bus. The Burj Al Arab is quite a distance (it is at Jumeraih Beach) but you do go near it on the Big Bus Tour.
In Oman we did the grand mosque tour which is excellent. In Abu Dhabi the first year we did Sheiks shopping but to be honest it is a bit of a rip off as there is a courtesy bus to the Marina Mall. In Fujeraih I wouldn't bother doing anything, have a relaxing day on the ship!
In Bahrain (we found it very dirty we did a tour again of a mosque and heritage centre. Didn't really enjoy that though. This year we took the courtesy bus and had a walk about but to be honest it is dirty and horrible so we went back to the ship.
If you want to see anything in particular the taxis are very inexpensive in Dubai.
Best ports are Dubai, Abu Dhabi and Muscat.