Well, You have two options here, first of them which I do recommend actually is to have an
Abu Dhabi tour from Dubai. Such a tour is available as bus tour of as private tour. bus tour could be at 200 AED.
Second option just in case you would love to do it on your own is to take a RTA bus from Dubai. There are two buses from two bus station in Alghubaiba and Ibn Batuta mall station. tickets is something like 30 AED per way. But in Abu Dhabi you need to hire private taxis to tour between attractions. Road between two cities could be like 1.2 Hrs if with a private transportation and for sure it takes more time if with bus tour.