Cotton should not be dry cleaned. The chemicals are too harsh for the cotton fibres. I learned this is one of the textile classes that I took when I became a seamstress. If you take anything cotton to a dry cleaner they will likely launder it and not dryclean it anyways.
As for the colors running, there are sheets that you can put in the washer with your colored quilt that the dye will adhere to and not run color into the other parts of the quilt. My friend washed a red and white quilt with these sheets and was impressed with the results. She did this in a front load washer.
My own method of washing quilts is to use the handwash cycle on my front load washer / cold water / light spin / lay flat to dry.
Lay flat to dry works well over the back of 4 chairs that are spread apart, then the air can get to your quilt from all sides.
