There are three ways that I know about, and as far as I know, there is no direct travel between the two places.
There are coaches (buses) from two different companies leaving the HK airport many times during the day that cross the border into Shenzhen where you may have to switch buses and then go directly to different places inside the city of Guangzhou, such as the China Hotel and the Garden Hotel. Inquire at one of the two desks in the HK airport about which destination in GZ will make it easiest and quickest to travel via another city bus, metro, or taxi to the GZ East railway station. This bus is approximately 200 RMB.
The second way would be to use local trains or buses in HK to go from the HK airport to the Hung Hom MTR station and take the Kowloon-Guangzhou Through Train (www.mtr.com.hk/throughtrain), which is a direct train to the GZ East station and costs about 200 RMB and there are only 10 of these each day. The KTT train running 3 times per day is preferred, but they're all good. There is an express MTR from the HK airport to the Kowloon MTR station for 90 HKD that takes about 20 minutes, as well as normal MTR trains that are less expensive.
Finally, there are local express city buses that takes a bit more than one hour to go from the HK airport to various places in HK island and Kowloon, including the Hung Hom MTR station. Bus A21 costs 33 HKD and goes to Hung Hom. You can also use the above to go from the HK airport and connect to the MTR East Line (it runs from Hung Hom to Lo Wu) that goes to the border crossing with Shenzhen at the Lo Wu station. Tung Chung - Lai King - Prince Edward - Kowloon Tong - Lo Wu in about 80 minutes. After you cross the border from HK Lo Wu over to Shenzhen Luo Hu station, you can take the CRH Guang-Shen Railway between Shenzhen and GZ East Station. They run every 10 minutes and cost 75 RMB for second class or 100 RMB for first. Second class is very nice; it's a modern, high-speed train put in about 3 years ago. I think the easiest way is to take the Airport Express MTR to the next station at Tsing Yi and change trains to the normal MTR Tung Chung line, then go two stations to Nam Cheong and change to the West Rail Line, then 3 stations to Hung Hom and take the Kowloon-Guangzhou Through Train for a direct and easy train to GZ East railway station. The only problem here is the limited number of through trains. The MTR website estimates 36 minutes for this trip from the airport to Hung Hom, but I think it will take longer. The benefit is that you don't have to ride a crowded local MTR with no seat for 30-40 minutes to get to the Lo Wu station. If you want to save a little money, you can use local trains to get to the Lo Wu station and the Shenzhen border crossing. From the airport, take the Express MTR to Tsing Yi and change trains to the normal MTR Tung Chung line, go one station to Lai King and change to the Tsuen Wan line, then 5 stations to Prince Edward and change to the Kwun Tong line, then go two stations and change to the East Line for your final trip north to Lo Wu. Be sure to use the correct train, there is another border crossing station.
(by courtesy of Mike, USA)