It seems a little bit complicated either by bus or subway. You need to connect at least one time in either way. So, maybe taking a taxi will be better for you.

hi, it is 7.1 kilometers from Guangzhou Railway Station to Guangzhou East Station. the most convenient way is to take a taxi, it takes around RMB25 for one way.
or you may take subway line 2 and connect line 1 to get there.

The following two ways are for your reference.

ONE: Take bus 271 (about 15 stops) at the opposite side of Guangzhou railway station, you will get to east station.

Two: Take subway line 2 (to Pazhou direction) and change to line 1 at Gongyuan qian station, the terminal is Guangzhou East railway station.