Get that walking stick out
Not so much Åre village itself, but the surrounding villages of Vålådalen, Storlien and others have great fell walking trails. You can walk for just a few kilometres up a fell or for days between fell cabins with your own tent. The choice is yours. A comment from me as a parent: children usually love this as they are completely free to roam and love adventures. Equipment depends on how long you will stay and walk but sensible shoes are a must as is mosquito repellant in July-August (although autumn is a nicer time with glowing yellow fellsides - in spring it is usually too wet after snow melting). A small first aid and emergency repair kit, a tent if you won't stay in fell cabins or stations (in stations they often sell dry and tinned food)...
Summer or winter - anything goes!
I cannot think of another Swedish place that has it ALL the way Åre does. In winter, it is a bustling ski resort, hosting World Cup downhill and other events (2007 World Championships) as well as playing host to loads of Stockholmers and others wanting to show off on the slopes. In summer, it has a different atmosphere altogether, as half the ski villages are closed and the village is calm and full of people enjoying a climb up the majestic Åreskutan or walks in the surroundings. The snowmobile tracks on the lake are gone and it is now glittering invitingly to motor boats, whilst on Renfjället above, reindeer go grazing, escaping the worst of mosquito pestering further down on the fell. Come here whatever the season and whatever your preference, and experience a place full of natural beauty, friendliness and good highland food.