One possible plan is like this:
First, get to Guangzhou East Railway Station. There are several ways, please refer to How to go to Guangzhou from Hong Kong. One good one is "Take the MTR to Lo Wu, pass the customs, and walk out and walk for 10-minute to the Shenzhen Railway Station. Then, take the train to Guangzhou East Railway station."
Then, take the No. 3 Subway (Metro) from Guangzhou East Railway station to the Baiyun Airport.
Note: there are two railway station in Guangzhou: the Guangzhou Railway Station and the Guangzhou East Rail station. You should get off at the Guangzhou East Railway Station to take the Subway to the airport.