There are mainly two means of transportation from Hong Kong International Airport, besides taxis: 1) Airport Express, the fast train from the airport to Central; 2) airport buses
The 1) option is expensive and serves only very limited areas. It is rarely the choice of local Hong Kongers. The airport buses with extensive connection to all corners of Hong Kong, with much cheaper fare, are the preferred choice.
The main airport bus routes start with A. Airport bus No. A11, for instance, goes to Central, Admiralty, Wanchai, Causeway Bay and North Point. Airport bus No. A21 goes through Tsim Sha Tsui, Jordan, Yau Ma Tei and Mongkok. All these tourist areas are well served by airport buses. Fares range between HK$30-40. Bus service is every 10-20 minutes.
For details of Route A11, see here.
For details of Route A21, see here.