Chaotic Hong Kong traffic and expensive taxis can mean that you're better off getting the train into the city.
Train
The Airport Express Line trains will take you to Central Hong Kong Station in 24 minutes, stopping only at Tsing Li and Kowloon Stations. Departures are every 10 minutes from 05:54-23:28 and every 12 minutes from 23:28-00.48.
There's a free check-in service in Hong Kong and Kowloon Stations to check in between 90 minutes and one day before your flight, and free porter services are also available. From Hong Kong or Kowloon Station, free shuttle buses run to many major hotels.
If you're going to Disneyland, you can change to the Tung Chung Line at Tsing Yi Station and travel to Sunny Bay Station for the Disneyland Resort Line. If you're going to the exhibition centre, AsiaWorld-Exp, take the five-minute journey in the other direction.
Tip: Download the 'MTR Next Train' app to find out the nearest departure times.
Cost: HKD 100 one-way to Hong Kong Station, HKD 5 to AsiaWorld-Expo
Location: Ground Transportation Centre between terminals
Taxi
A taxi is a practical option, but is not cheap. Hong Kong taxis are colour-coded according to their operating areas. Urban taxis (Red) serve all destinations throughout Hong Kong including the airport except Tung Chung Road and roads in south Lantau. New Territories taxis (Green) serve only the New Territories and specific roads in Lantau. Lantau taxis (Blue) serve all destinations in Lantau and the airport.
Cost: Around HKD 300, plus HKD for each piece of luggage
Location: Outside Arrivals, to the left
Bus
To save money, there are many, cheaper, buses taking you to many destinations in the Hong Kong area. For central Hong Kong, the S1 Citybus to Tung Chung MTR Station and board the MTR Tung Chung Line, taking you along the same route as the Airport Express for less money, but stopping at more places. For more on bus times and routes, check out: www.citybus.com.hk or Long Win Bus at www.lwb.hk.
Cost: Fares vary but are in the region of HKD 30-50; the S1 costs HKD 3.5 to Tung Chung
Location: Ground Transportation Centre between terminals
Ferry
You can take a ferry to other destinations in the Pearl River Delta from the airport's SkyPier. Access is before immigration, as you're not considered to have entered Hong Kong at this point. Destinations include: Shekou, Shenzhen Airport, Macau, Humen, Guangzhou, Zhongshan, Zhuhai and Nansha.