Most of the settlements are linked by Grønlandsfly (Greenlandair), Greenland's national airline, but the weather can always play havoc with the best of the airline's intentions. The airline has sometimes been referred to as Immaqa Air, loosely translated as Maybe Air, and it's wise to bear this in mind when booking. Leave plenty of leeway to account for grounded planes and other weather-induced delays. Planes may also be chartered from Greenlandair; most of them are five-seaters.
Travelling by boat is a good option; a fleet of coastal ferries run up and down the west coat from Aappilattoq in the south to Uummannaq in the north. They're not the QEII by any stretch of the imagination, but they're safe and go where the Queen's ship would fear to tread. The exception to this is the grand old dowager, the M/S Disko, which has been upgraded and been pressed into service as a tourist cruise ship on five-day or eight-day cruises. For those inaccessible villages and towns, or places off the usual ferry route, charter boats are available.
If you're thinking car or motorcycle, forget about it. Glaciers and ravines tend to make road-making a thankless task in Greenland and there are only two settlements connected by tarmac; Ivittuut and Kangilinnguit. You can pootle around the settlements if you really feel like driving - most Greenlanders do - but the 4WD hasn't yet been made that can cope with Greenland's rugged and icy conditions. Dogs, on the other hand, are capable of going anywhere, anytime, although legally dogsleds are only allowed in towns north of the Arctic Circle.