What is the convenient way to travel from Qingdao to Zhangjiajie?

First of all, I'd like to tell you that no direct trains run for the above journey, so you need transfer indeed. The reommended transfer stations are Zhuzhou or Yingtan. Anyway, it is K train that takes you for the whole journey.
Also there is no direct flight between the two places, you can take Changsha as the transfer place.
Actually, it will be a little cold and rainy in Feb. So I suggest you go there in Mar. or Apr.