There are regular buses (every 15mins or so) on the E1 route between Dubai and Abu Dhabi, the bus station in Dubai is in Bur Dubai, the centre of the old town, near the creek, and the bus station in Abu Dhabi is in Al Wahda on the Muroor Road (East Road), close to the centre of the city. The cost is Dh20 from Dubai to Abu Dhabi and Dh15 from AD to Dubai (a taxi will cost you around Dh200). The buses are big, air-conditioned and have plenty of storage space for luggage. In Dubai, buy a ticket from the ticket office; in AD just find the bus stand (near the main entrance off the Muroor Road) and pay as you get on the bus.