Entering HK from Guangzhou

I think you misunderstand the 144. The 144-hour visa free transit is only for transit passengers in Mainland China, and HK is not included. You can go directly to HK from Mainland China without visa as you enjoy 90 day visa-free entry to HK. And I can tell your route is Republic of Korea - Guangzhou - HK. You can enjoy the 144 to stay in Guangzhou for 6 days. And you can enter HK from Guangzhou by high speed train.

Yes, you are allowed to exit Guangdong province by high speed train to Hong Kong. This is a brand new addition to the number of eligible exit ports along with Pazhou ferry port located practically in the center of Guangzhou. You should just have in mind that you have to book your high speed train ticket in advance, even before your departure for Guangzhou, and it is highly advisable to print it out and to have it at hand, ready to be presented to relevant authorities. Also, although I assume that you are informed about the details, you have to arrive into Guangzhou by air, but from a third country different than Hong Kong and you are not allowed to have any additional landing immediately before Guangzhou in any other mainland Chinese airport. In other words, it is fine to have an itinerary like Seoul - Guangzhou - Hong Kong, but not Seoul - Shanghai - Guangzhou - Hong Kong.

Also, you have to know that you can return, if you plan that, from Hong Kong to Guangzhou by train or any other option beside taking a plane. So, entries can be done by air only, but exits can be performed in many different ways (through many different ports).