The bus system is super easy to figure out. If you want to go to Epcot, you get on the bus that says Epcot. If you want to go to Animal kingdom, you get on the bus that says Animal Kingdom. Simple as that.
One thing to keep in mind, though, is that there isn't direct transportation between the parks and Downtown Disney. You would need to transfer at a resort.
Except that the OP also specified some park-to-park transport routes, which isn't necessarily as clean as that. You can't "get on the bus that says Epcot" at the MK (or vice versa), for example, because there won't be one.
Here's the skinny:
Resorts to Parks (and vice versa)
Direct or shared buses are available from all resorts to all parks,
except when going from the CR, Poly or GF to the MK or from S/D, BW or YC/BC to Epcot or DHS. In the former case, take the monorail or boat between the Poly or GF and the MK, or take the monorail or walk to/from the CR. In the latter, you would walk or take the boats.
FW and WL guests can take bus or a boat to the MK.
Park-to-Park
Direct bus service is available between AK and Epcot, AK and DHS, and DHS and Epcot (main gate). Buses also run between AK and DHS and the MK, but the MK stop is at the TTC and requires transfer to the monorail or ferry to get to or from the MK to the buses. The Epcot-MK trip is made via monoral between Epcot and the TTC, and via monorail or ferry between the MK and the TTC.
There also is boat service between DHS and Epcot (International Gateway) or you can walk from one to the other via BW.
Resort-to-Resort
The trickiest trips using WDW transport involve going from one resort to another. Other than a few exceptions, there is no direct transportation service.
Obviously you can get between the GF, CR and Poly on the monorail and you can walk or take the boat between the Epcot area resorts (BW, S/D and YC/BC). There's also boat service between FW/WL and the CR.
For the most part, however, you need to use a park (as long as it's open) or DTD as a transfer point to get from one resort to another. DTD generally is your best option only if all of the parks are closed OR if you are going to or from SSR and are willing to walk between that resort and DTD.