There is no airport to Eagle River, the train does not stop in the town either. To go to Eagle River, you must either drive or take the city bus "People Mover" from Anchorage. The bus system can pick you up at the airport, for schedules and information please see http://www.peoplemover.org
This is a suburb of Anchorage. Many people live here and commute to Anchorage everyday, adds about an hour of driving each day to your life. There are no airports here, you have to drive 45 mins to the Anchorage Airport.
Eagle River is bike friendly, the commercial district is very compact. Hope your leg muscles are in top shape, this town is built on a valley and therefore has many steep inclines!
There is also a bus route into Anchorage, and many people bring MotorHomes here and camp for the summer.