By ship
Most cities and villages in the area are situated by or near the fjords, and local travel is therefore mostly by boats. The tourist offices in the cities arrange local boat transport, but you may be able to hitch a ride on a private boat. No matter how you do it, you will be sailing in beautiful clear water in scenic fjords with icebergs floating around you. Don't worry about the icebergs; the locals have been sailing the waters all their life and know not to sail in bad weather or too close to the icebergs.
On Land
Some of the villages, like Narsaq and Qassiarsuk are connected by roads, and the locals travel around on farming equipment, jeeps or 4 wheeled scooters. Some places you may be able to rent one at the tourist office.
Hiking is quite possible and hiking maps can be bought at scanmaps (http://www.scanmaps.com), but beware the low scale, and the large difference between contour lines, making precise navigation difficult. Also note that tour suggestions merely means you might be able to follow the route, as no trails are marked and streams and lakes may vary in size throughout the year.
On a lot of the islands and peninsulas are sheepfarms. Aside form meeting an occasional flock of sheep, this has two important side effects:
The sheep keep away the mosquitos!, making the stay much more pleasant.
The sheep are lazy, and take the easiest route from A to B. If you are lazy too i.e. not a rock climber use the paths the sheep have already made. Just remember that your destination is not always the same as the sheep's.