Worked there in the late 90's before the Palm & Burj Khalifa went up & there was none of the attractions that bring visitors flocking in these days but the highlights for me, back then, would have been walking around the creek area and watching the dhows loading and unloading and then heading down to the gold souk. Taking an abra across to the other side always felt like an authentic Dubai experience & there are many restaurants worth exploring on the waterside.
If you hire a car, can definitely recommend driving across to Fujeirah on the East coast and stopping off at Hatta on the way for lunch at the Hatta Fort Hotel. I much preferred the beaches on that side as, in Dubai, the beaches shelve very gently so the water is shallow for a long way out & is like entering a warm bath in the middle of summer. Not really what you want, though I can only talk about the public beaches I used to visit; there may be better ones where the hotels are located. In fact, proably best to avoid public beaches, if you are female, as you are likely to get unwanted attention.
EDIT: Just seen OP's response above