Taxis are abundant in Hong Kong. In most city areas, you'll find just the regular, red taxis; in the mainland New Territories there are green taxis, while blue taxis serve most of Lantau Island.
Red taxis are priciest; flag falls and per distance rates for the green and blue taxis are lower, as they tend to cover longer distances. There may be extra charges, for bags, tunnel fees, phoning for taxis.
In the main downtown areas, you can flag down taxis in many places - but not where there's a fence alongside the road. Scattered near the harbour, there are ranks for cross-harbour taxis: Hong Kong Island taxis that have wound up on Kowloon side, and vice versa: take one of these, and you only have to pay a one-way tunnel fee, as the driver returns to his (most drivers are guys) area.
HK New Territories and Lantau taxis
Once you're in the New Territories, you can save money by taking green rather than red taxis (though you'd need a red one if you're heading to or south of Sha Tin/Kowloon, where green taxis aren't allowed [there are interchanges, for journeys starting in green, transferring to red). In some places, like near Tsuen Wan station, you'll find there are both green and red taxi ranks, so can choose green if you're going somewhere rural.
Even once you seem to be right out in the boondocks, if you're near a road there's a fair chance a taxi won't be long in coming by. There are a few exceptions, where if you really want a taxi you should perhaps call for one: for me, an example is Mai Po, where can call 26778888 - which may work in plenty of other parts of the New Territories.
On Lantau, the blue taxis are relatively scarce, but still convenient for roads south of Tung Chung (and helping to link the airport to the south of Lantau).