You will set the output of your video card to match the maximum resolution your TV can play. For most people (in the US), this will be 1080p 60Hz. It will no longer matter what the input resolution of your cable channel is, the video card will change it to be 1080p 60Hz and send it to your TV that way. If the channel is 720p, your video card will change it to 1080p and send it out. If your channel is 1080i 30Hz, the video card will change it to 1080p 60Hz and send it out. The higher the resolution of the original channel the better - but all the non-HD channels will be converted to 1080p and sent out that way.
The nice thing about this is you do not have to worry about any of the other components in the system needing to perform HDMI handshakes, etc., each time the resolution changes - since it will never change for you.