Despite their close proximity, traveling between Greenland and the rest of North America is not an easy (or cheap) task. Currently (January 2012), there are no commercial flights between the US/Canada and Greenland. Flights into or out of Greenland are only offered via Denmark or Iceland.
Greenland's domestic air carrier, Air Greenland, basically has a monopoly on all flight travel in Greenland, as evidenced by its high flight prices. Air Greenland offers regular flights from the major towns on the west coast of Greenland (including Nuuk) to Copenhagen with stops in Kangerlussuaq. The airport in Kangerlussuaq (an old US military base) is the only operating runway in Greenland large enough for commercial jets. All other flights essentially shuttle passengers to Kangerlussuaq via smaller Dash-7 or Dash-8 prop planes. From Kangerlussuaq, an Airbus flys daily to and from Copenhagen.
That's it for airline options, and whether you fly via Iceland or Denmark, connection layovers are long. Normally, an overnight stay is required in Copenhagen. If flying via Iceland, plan on spending most of the day in Reykjavik (not necessarily a bad thing--we suggest you check out the municipal swimming pools).
To top it all off, checked baggage on both Air Greenland and Air Iceland is limited to 20 kg, and excess is charged the ridiculous rate of about 10 USD/pound. Often times, this is ignored (at least with Air Iceland), but it's hard to know. As an example, we've decided it's cheaper to just buy new skis in the US rather than pay the fee to carry ours back and forth with us.
Because most of these airlines don't partner with major travel sites like Kayak, Expedia, etc., it's difficult to create a single itinerary. Therefore, it's a good idea to find travel insurance that will cover cancellations, delays, etc. for one trip booked on separate itineraries on multiple air carriers that aren't partners. Any suggestions from the seasoned travelers out there?
Lastly, be warned-- winter travel is especially risky. Because of Nuuk's short runway and dodgy weather (not just snow/ice, but also wind), delays are common. I've been told that to fly between Nuuk and Reykjavik, airlines require that Kulusuk airport in in East Greenland also be open as a backup landing runway before they will make the trip, but we're not sure what this consists of-- possibly just a guy with a radio who feels like getting up that day and saying ,"All clear!".
As you might imagine, the simultaneous occurrence of good winter weather in Iceland, West Greenland, and East Greenland is fairly rare, so windows for travel are small. On my travel to the US this winter, I was delayed 5 days. On my way back home to Nuuk, I spent 4 days holed up in a hotel in Reykjavik because of weather delays. It's all a bit demoralizing, but better than using your seat cushions as a down sweater in the even of a crash on the ice cap, as recommended in the Dash-8's emergency safety brochures.