A sledge dog ride is surely one of the must does, if you are visiting Greenland north of the polar circle. The legislation in Greenland stats that sledge dogs are the only dog bread allowed north of the polar circle and to the contraire disallowed south of the circle.
This activity is naturally depending on the ice/snow cower you can assume the winter season will give you rich opportunities but that it is most enjoyable in the late part of winter April - May as the temperature and daylight is on the rice.
The going rate for a trip for an hours time is 600 DDK (100$) pr person.
In Aasiaat I can recommend Lars Iversen he is the local champion in dog sledge racing (2011) out of the last 5 years he won 4 times, he only speaks Greenlandic and Danish, so you might need some help with arranging the ride. Phone +299 291519
There is very fine opportunities for fishing in Aasiaat, for example on my first fishing trip I did catch 9 codfish (15 kg in all) in 3 hours from the cost. I used artificial lures and snood with some roast from last days dinner as bait (is worked very fine).
If you are in town for a visit there is only one way to get from the airport to town and you might also like to take a taxi around town. A trip from the airport to the town center will cost you around 110 DDK (20$).
You will have to call the taxi company to get a cap to come and pick you up.