The walking path to the contemporary - after you go through security at MK, take a right for the path CR . You have to cross a street closer to the hotel but there’s a light and cross walk. If you want to take the monorail- get on the Resort monorail not the express one. The Contemporary is first stop after magic kingdom.
If you’re not going to a park after eating, You can take the bus to Hollywood Studios. Bus stops are on first floor. Then Skyliner to CBR and then Pop Skyliner line or just walk back to MK and get Pop bus. The monorail is going to stop at the other hotels before MK so I would not recommend it for journey back unless you want a scenic route