Hong Kong Railways
Trains to Hong Kong's New Territories and Lantau Island
Hong Kong's major rail network is operated by the MTR (Mass Transit Railway) Corporation. It includes a system akin to the London Underground or a subway - with mostly underground lines; as well as a railway line to Tung Chung and the airport, north Lantau, and the north-south running East Rail and West Rail. The latter two were originally operated by the Kowloon Canton Railway (KCR), but this was merged with the MTR Corporation in December 2007. There's also a Light Rail system in northwest Hong Kong.
There have been recent changes to the system, including the addition of a branch line from Tai Wai (on East Rail) to Ma On Shan, and a southward extension of West Rail, to reach Hung Hom.
Subway: the classic MTR
The subway - the "classic" MTR - is wonderful for getting around in the city, with lines serving much of north Hong Kong Island, as well as Kowloon and, via the Airport Express line, north Lantau.
There are interchanges with other lines, including with East Rail at Kowloon Tong; and with West Rail at Nam Cheong, Mei Foo and Tsim Sha Tsui.
The classic MTR's Diamond Hill and Choi Hung stations are close to stops for buses heading to Sai Kung and Clearwater Bay; Shau Kei Wan station is by the bus station serving Shek O. Other stops that are useful for reaching Hong Kong's outdoors include Wanchai (for nearby hillside), Wong Tai Sin (for Kowloon Hills) and Central (particularly as there are ferries to islands, and buses to destinations such as Stanley and the Peak).
East Rail: towards many places in the New Territories
Cross-border trains run on East Rail, but most trains are local, running between East Tsim Sha Tsui in Kowloon (the station here has underground walkway to "MTR" line - takes maybe 5 minutes or more, so if I'm northbound from Central I still keep to MTR to Kowloon Tong, then change), and stations to Lo Wu/Lok Ma Chau on the border.
Tai Po Market is among key stations for accessing rural Hong Kong: from here there are buses east to Tai Mei Tuk and Bride's Pool, and west/orthwest through the Lam Tsuen Valley with stops at or near Kadoorie Farm and Ng Tung Chai; and you can take a taxi to Tai Po Kau Forest Reserve. University station is a roughly ten-minute walk from the Ma Liu Shui ferry pier, from which there are ferries east through the Tolo Channel.
Especially if you ride a train to/from the border at Lo Wu, East Rail can be pretty crowded, so you may want to consider riding First Class. If you do so and you're using an Octopus card, remember to wave it over the upgrade terminus-thing, on a wall or pillar by the entrance for the First-Class compartment. (Yellow signs point the way to the spot on the platform where the First-Class coach will halt.) I forgot once - and, oh dear, led to prompt HK$500 fine!
West Rail offers a fast, smooth ride, but unless you are interested in birding or culture it isn't so great for getting to wild Hong Kong. This is because, the line mainly serves northwest Hong Kong - much of which has been trashed by container parks, scrapyards, and development that generally seems to have occurred with little or no sense of aesthetics. But you may have your carriage almost to yourself. (Even people commuting from the northwest aren't flocking to West Rail.)
The Kam Sheung Road station is within a relatively short taxi ride of Mai Po and close to Kam Tin; Tin Shui Wai is beside the Ping Shan Heritage Trail, close to Tsim Bei Tsui (a taxi ride), and a Light Rail trip from the Wetland Park.
The Light Rail system connects urban areas in northwest Hong Kong; there's a station by the Hong Kong Wetland Park, Tin Shui Wai.
The Peak Tram is a venerable and famed funicular, climbing the hillside above Admiralty (near Central) to a stop high on the Peak. Very useful for getting to the start of walks on and from the Peak; though there are sometimes large crowds - in which case you might find it quicker to take a bus.