By plane
It is a 40-minute bus ride from Fukushima Airport to Koriyama station ¥800, from which you will then have to travel to Fukushima station by Shinkansen ¥2920, 15 minutes or local train ¥820, 45 minutes. It may be a bit more convenient to arrive in Tokyo and take the Shinkansen from there.
By train
Fukushima is a stop on the Tohoku Shinkansen line between Tokyo and Sendai, and is also served by the JR Tohoku and Yamagata lines. The Yamagata Shinkansen branches off here, traveling over conventional lines to Yamagata and Shinjo.
The travel time from Tokyo to Fukushima by Shinkansen is approximately 1 hour 45 minutes, with 2 to 4 departures every hour at a cost of ¥8500 each way non-reserved seats.
By bus
JR Bus' Abukuma makes five daily round-trips to Fukushima from Shinjuku Station in Tokyo 5 hours, ¥4800. At night, JR Bus runs the Dream Fukushima from Tokyo and Yokohama stations 6 hours from Tokyo, ¥4800. Tohoku Bus' overnight bus, called the Suite, is cheaper at ¥4500 but arrives earlier in the morning.
Kintetsu Bus and Fukushima Kotsu's Galaxy bus runs overnight to Fukushima from Kyoto 11 hours, ¥12130 and Osaka 12 hours, ¥12740.
Willer Express (http://willerexpress.com/) offers bus service to Fukushima from Tokyo, Nagoya, Kyoto and Osaka, with online bookings available in English. Willer Express' prices vary by the day you are traveling and the class of bus. Willer Express offers "Standard" similar to Sakura's offering, and "Relax" slightly larger seats with more leg room, a blanket, and a dome looking thing to keep the light from your eyes.