When you connect to composite video to a shared component-composite video input you will get a bw picture if you don't go into the TV menu and set the input to composite. Do that and you should be fine.
You will also need to connect a stereo RCA audio cable from the dish box to the TV to get sound.
If your new TV is HD it will be worth getting an HD Dish box and connecting it with HDMI. HDMI has digital video and audio on one cable. The difference in picture quality will be huge.