Is it possible to take a train from Kowloon to Yangjiang? If not, is there a bus?

I'm afraid no direct train run between the two places. And I don't find the bus schedule either. So I suggest you take the cross-border coach at Mong Kok or Jordan to reach Guangzhou first. You will drop off at Hotel Canton, Guangzhou Hotel or other hotel. After that, you can take a taxi to reach the local Provincial Bus Station (near Guangzhou Railway Station), where you can catch the direct bus to your destination.

Or you can take East Rail Line at Hung Hom, Hong Kong to Lo Wu and cross the border to reach Luo Hu, Shenzhen. After that, you can walk to the nearby Qiaoshe Bus Station, where there are direct bus to your destination city.