Hong Kong - Train System
Since Hong Kong is a city, and a Special Administrative region of China which requires a visa to enter the rest of China, there is no where to go within Hong Kong by train.
Travel by train is not very common unless you are going to Shenzen, China for the day. You can ride the Kowloon Canton Railway (KCR) which connects to the MTR at Kowloon Tong. This line will take you up to the boarder with Shenzen, but make sure you have your passport and China visa before you arrive!
MTR - Mass Transit Railway
The MTR is clean, fast, safe, and a great way to get around. There are a number of lines so the North side of Hong Kong Island and most of Kowloon are accessible by MTR. The sign for the MTR is a red circle with a white insignia. If you go into an MTR station you will find maps of the whole system. You can pay by buying an individual ticket, or use the Octopus card. More information, including a map of the MTR and information about the Octopus card, can be found at www.mtr.com.hk You will also be able to see the sign for the MTR so you can recognize it.