My two cents!
I lived in Anchorage for 17 years and just moved in the past couple of years to Colorado. So anyways, 1st bit of advice, don't go in the winter, go in the summer, you'll enjoy it a million times more. Now that that is out of the way, here are some winter attractions that you'll love.
Restaurants:
Anchorage-
*Kaladi Brothers Coffee Co. (any coffee you get you must get it from Kaladi's, great roasts and awesome people.)
*The Moose's Tooth (great pizza, fish tacos, etc.)
*The Bear's Tooth (same company as Moose's Tooth, expanded menu, and also has a dine-in movie theater. Moose's Tooth Brewing Co. makes some outstanding beers.)
*Orso's (little fancier dining)
*Snowgoose Cafe
Girdwood (30 miles south of Anchorage)
*Double Musky (rated as one of the best restaurants in the whole nation)
*The Bake Shop (go in before lunch for a great breakfast)
Things to do:
*Ski/Snowboard at Mt. Alyeska (in Girdwood, stay at the resort too, you will not regret it. They have a lot of activities down there as well.)
*Sled dog rides/tours (look online or phone book when you get into town.)
*Watch the Northern Lights (Aurora Borealis) if it's a clear night.
*Snowmachine Tours
*Ice fishing
*Hike "Flat Top" but if it is cold, that's a very long and tiresome hike if there is fresh snow.
*Cross-country ski. Endless miles of trails in the Anchorage bowl.
So in the winter there is not a lot of other indoor recreation, but if you want to stay warm, take some drives (Rent a SUV or AWD) north and south of town and enjoy the magnificent scenery.
Any ocean activities are out of the question because it will most likely be iced over (unless you love scuba diving in 40 degree water.)
I'm trying to think what else. In the winters everyone goes into their own little hibernation and unless you love the outdoors, you usually do indoor activities. If I think of other suggestions, I'll add!
Ryan Walker