We flew Delta from Florida. No direct routes but never had any problems either with connecting flights, etc. Big Mountain offers a shuttle from the airport to their hotels on the mountain.
In the winter if you stay on Big Mountain you can definitely utilize the SNOW bus. It got a little crowded towards dinner time in the evening but was good on the whole. We stayed there 2 weeks, no prob getting into to town for dinner or shopping.
You can fly to Kalispell, 15 miles away, take Amtrack right into Whitefish, or drive. I guess you can get there by bus, but i never investigated that option.
The local bus system is reportedly very reliable--the SNOW Bus. It will get you around town and up to the mountain for skiing.