Dubai is such a big city that you will need time to cover its main attractions, However let me help by sugessting some suggestion that you may avail to get around the big hub.
You firstly may try the big bus tours, it takes you to the may attractions of the city in a quick tour around. the tours for sure are shared ones. the bus itself goes from certain points at certain times, you may need to check their website to identify them.
Another sugesstion here, is the Dubai bus tours which are operated by local tour operators. it is the same purpose of the big bus tours but this time is operated by locals. the tour extends to like 4-5 hours and takes you also to the main attractions of city just like palm jumierah, burj arab hotel, old gold and spices souks, jumierah mosque and the city museum.
I believe it may be the best way to discover the city specially during day time, the bus tours also are very budget to take specially because the locations where the bus goes are not near to each other, So if you did on your own you will spend so much time and money on taxis and so on.
However, there are some attractions I recommend you to do on your own, just like Dubai mall complex which includes Dubai mall, burj khalifa and dubai fountain. All you need to use metro to get to the station which called after the mall; Dubai mall and from there you could go into the mall and exit to the group floor to watch out the fountain shows and checking burj khalifa. even if you needed to go up at burj khalifa you could use their online website to book tickets online. So I see that hiring a tour company for such a visit is really a waste of money.
Also you could head to the miracle garden using metro and bus, global village using bus, mall of emirates usuing metro and so on.
One important notice here is that you need to buy and charge your metro card once in Dubai so you could be able to use it.