Kodiak is serviced from Anchorage by ERA Aviation and by Alaska Air. Era operates as a subservice of Alaska, so all arrangements go through Alaska Air.
About 8 flights a day - 6 by ERA's Dash 8 turbo prob and 2 by Alaska jet. About 45 minutes by jet or 1:10 by turbo prop, but the turbo is really faster because you don't have to deal with security/TSA hassles. So, if I'm just going to Anchorage I'll usually stick with ERA. If I'm going beyond Anchorage I'll take the jet and then I'm already thru TSA in Anchorage.
As of 6/1/09, airfare is about $500 round trip, but web specials for $199 can sometime be found at www.flyera.com. I know that seems steep, but after 18 years of paying $900 to get to and from Dutch Harbor/Unalaska it doesn't seem that bad.
Be sure to sit on the right side of the aircraft coming into Kodiak for the best view as the flights pass by the town of Kodiak on final approach to the airport.
You have to fly to get there quick, take the ferry if you have some time.
Era Aviation (www.eraaviation.com) and Alaska Airlines (www.alaskaair.com) both fly in many times a day.
You should rent a car, there are car rental agencies located in the airport.
In Alaska we rely on air charters to get around to remote villages, or just to get off the island. Island Air has service to the native villages, as well as the smaller water stops, places without a runway.