Two options from Guangzhou airport to Changping:
Option 1, take airport express bus (Line 6A) to Zhongxin Mansion, about 45 minutes. Then walk through the square (about 200 meters for 5 minutes) to enter Guangzhou East Railway Station, where you can buy high speed train ticket from to reach Dongguan Station (not the east one). The station is in Changping district.
Option 2: At the exit of the airport terminal, there is a ticket office selling bus tickets to every city temrinals. You can take the bus to Dongguan (Dalang Town), about 100 minutes. Then take local cuty bus to Changping, 30 minutes. Or take taxi, for 40RMB which is faster. No direct bus from Changping to GZ Airport, unless you go to the city terminal in Dalang.
Hope this is useful!