How can you get from Epcot to Downtown Disney and then from Downtown Disney to MGM. Can you do that using the buses?
There's no direct Disney transportation between DTD and any of the theme parks. There used to be, but a lot of folks took advantage of the free parking at DTD to avoid paying to park at the theme parks (they'd drive to DTD and then take a bus to a park from there).
To get to DTD via Disney Transport, you have to get there from a resort. There are direct buses from each resort, as well as boats from Port Orleans Riverside, Port Orleans French Quarter, Saratoga Springs Resort or Old Key West.
So from Epcot you could walk out the International Gateway, walk to the nearest resort in that area, and hop a bus to DTD. Or you could go out the front entrance, hop a bus for POR, POFQ, SSR or OKW (since they are near DTD and are on the way from Epcot), and from there hop a bus OR boat to DTD. If you go to SSR, you can also walk to DTD from there (although I'm not sure which stop within SSR is the closest to DTD).
From MGM, pretty much same as above as with Epcot.
A taxi would be the quickest way, but then again it's not free.