Choose a medium-to-light shade of butter yellow as the main paint for the living room. Accent it with white trim and decorate with dark furniture upholstered in brown, dark blue or another dark shade. Brighten the look with a white throw on the couch and pillows in a print that has a related shade of yellow.
Paint the dining room, which opens off the living room, the same shade of white you used on the living room trim or a very pale shade of butter yellow. Use a more vivid yellow on an accent wall, which can be seen from the living room. Use more of your white, pale yellow or one of the darks from the living room print for accessories, seat cushions and a table runner. Using shades of your main color and small touches of your living room accent colors encourages the eye to see the two rooms as part of a larger whole.
Turn the color tables a bit for the family room, which can also be seen from the living room. Consider a dark neutral such as stormy gray for the walls, upholster the love seat in a gray and yellow check, and add plain yellow throw pillows. Furniture can be either dark or light, curtains can echo the pale yellow used elsewhere, and a large mirror correctly placed reflects parts of the other two rooms.