Yes there are tour groups with different packages. For the Desert Safari you will need a tour operator anyway. You can’t go there on your own. The operator I have used twice was FSN Travels. First time around they were great, but on the second desert safari they were extremely late in picking us up, so by the time we got to the desert it was dark already.
As far as the city is concerned I think it’s good to explore it yourself, if you have enough time.
If you don’t, then the Big Bus Tour is a good way of seeing much of Dubai in a short period of time. You can hop-on and hop-off at different places. A friend of mine did that recently and he really enjoyed it.
Here’s the link: Big Bus - Dubai Sightseeing Tours