The best (quickest) way is to drive, but I will assume that you will not have a car and want to use Disney transportation.
If the MK is still open, take a bus from POFQ to the MK. From there, you can either walk (about 10 minutes) to the Contemporary or you can take the monorail. The walk from the bus stop to the monorail is only a little shorter than the walk from the bus stop to the Contemporary, but IMHO, the monorail is more fun! After dinner, you can just take a bus to DD/PI.
If you are already at DD and want to go to Chef Mickey's from there, just take the bus to the Contemporary.
If the MK is closed when you want to go from the POFQ to the Contemporary, I would take a cab. The other methods will take quite some time. If time is not an issue, you can either take a bus from POFQ to DD and then transfer to the COntemporary bus or you can take a bus to EPCOT. Then take the monorail to the TTC and transfer to the resort monorail.
HTH.