To get to the mountain follow the Appalachain Trail! Katahdin is just north of Millinocket, ME, in Baxter State Park. There are detailed directions and insructions for winter climbing on the state park web site: www.baxterstateparkauthority.com/ The park is awesome in the winter and offers full bore, hard alpine climbing as well as shorter, more friendly ice climbs. Expect to spend two days coming in and out as you are required to stay at the Roaring Brook Campground on your first and last night in the park.
Mountain Project's determination of some of the classic, most popular, highest rated routes for Katahdin:
A long, great gully climb. When I climbed it there was very little ice, making it a snow climb with a few easy rock bands. The hardest part for us was actually getting onto the ridge at the top of the climb to head out for the knife edge and the rim of the basin, though by heading left down the Dudley trail this crux may be avoided. DO NOT CLIMB WHEN THERE IS AVALANCHE DANGER!...[more]Browse More Classics in ME
In the summer the park rangers will hand out a "Technical Rock Climbing At Katahdin" paper which has general information (basic route descriptions with grades that are not all that helpful.) There is also an informal guide book with photographs and drawn lines of the routes along with descriptions from many of the first ascentionist. Be prepared to show your gear and lay out your intentions to the ranger on duty, who may or may not know something about climbing and may or may not be polite and respectful.