The main mode of transportation here in Dubai is still the Public Buses. And RTA or Roads & Transport Authority is the only authorized (or you can say monopolized) public transportation provider. Not that I complain of, because they really provide top notched buses (look at the pictures) and courteous drivers (maybe not all). They also use an automated fare collection, so even if you use the bus, train or the water bus, you will be charged for the distance and not by the times you use the service.
Sample RTA buses, the double-decker is usually used for inter-emirate. I already rode one going to Sharjah.
There are several bus stations and stands (pick and drop) around the city. You just have to memorize the bus numbers and routes, which I am not very good since I seldom ride buses (I am more of a Taxi lover). I only know 1 bus number going to our place. But if you have difficulty like I do, you can use the online Journey Planner.