There are buses from park to park, except to Magic Kingdom. Park buses leave from the TTC, which is across the Seven Seas Lagoon from MK. So you would first take the Express monorail from MK to TTC, then switch there to the MGM bus.
It might be quicker to take the Boardwalk resort bus from the Magic Kingdom, then walk to MGM via the 3/4 mile path that begins at the rear of the Boardwalk. You can also take the Friendship boat from the Boardwalk (or Swan/Dolphin) to MGM.
It is possible to go from park to park on WDW transportation, but it always involves some sort of transfer. As noted by previous posters, either of those methods would be the fastest to get to/from MGM from the MK.