There is direct transportation from each park to all three other parks; however, the park-to-park transportation at MK is actually at the TTC, not at the park gates.
From MK to Epcot, there is a monorail line. From MK to DHS and AK, there are buses.
From Epcot to DHS, there is a bus at the front entrance, and boats from International Gateway. From Epcot to AK, there is a bus.
From DHS to MK and AK, there are buses. From DHS to Epcot, there is a bus to the front of Epcot and boats to International Gateway.
From AK to all parks, there are buses.
There are no buses from any of the parks to Downtown Disney, or vice versa. To reach DTD from one of the parks, you need to go to a resort first, then transfer to a bus; all resorts have buses to DTD. The two Port Orleans resorts and Saratoga Springs also have boats to DTD, but the boats only run in the evenings after 4pm.
From personal experience, I can tell you that driving yourself is faster than Disney transportation in almost every case. It's a little slower when going to MK because the MK lot is on the far side of the Seven Seas Lagoon.