Bus from Art of Animation resort to Magic Kingdom, walk (about 10 minutes from MK bus platforms) to Contemporary Resort.
Note that you could also take the monorail from MK to CR but the walk from the bus platform to the monorail platform is about half the distance of bus platform to CR and is off in the opposite direction geographically.
Nitpicking: If it is pouring rain you might try AoA to Hollywood Studios and take the CR bus from there, since the walk between bus platforms is much shorter, something like under one minute.
With the exception of bus routes (or the monorail route) that serve more than one resort, all trips from one resort to another require making a connection.