How to get better picture quality out of my LCD TV?

You are using a composite cable, the Yellow for the video and the red and white for the left and right audio. This cable is only capable of the 280P resolution. At the minimum your tv will have a 720P resolution. You need to use either the component cable (Red, Green and Blue rca jacks) for the video and optical audio cable for the audio or the HDMI cable that can handle the 1080P and 7.1 audio in one cable. Video and audio lag is due to having a cheap bargain price tv with a slower response time. Don't confuse this with the refresh rate, which is due to blur motion when watching fast action sports or video games. You also need to configure your xbox to the right cable you are using. Hope this will help you out.

I think you need to buy an HDMI cable and everything should be good after that. But i honestly am not sure about the input lag. But the HDMI cable will make everything look a lot better and could possibly fix your lag. Go to Best Buy and get one, buy the cheapest one they have. They will try to tell you that the higher priced wires are better but they really aren't.

What colors are the cables? It seems you are using regular RCA yellow, red, white which are standard definition. You need to use either component cables red, green, blue or a HDMI cable. I've seen some xbox that have a switch where the cables are to change from standard to HD. Make sure the switch is set to HD