Animation is one of the things that Maya does best so there is not really any reason to go to another app to animate, however once you have your animation perfected in Maya i would not save the file as a movie format i.e AVI but render out an image sequence instead and use something like After effects or another app to create your movie file.
Incidentally although it is possible to do some animation in AE its not really an animation app but more a compositing platform with powerful effects tools for things like colour correction ect.