How can I connect my laptop to my TV without using my TV's VGA input?

Be very careful; Do not purchase a VGA->Component cable, it will not work. These cables are for converting Component-over-VGA to component cables, for devices that already output a component signal to a VGA port. Your laptop does not do this, and you will need an actual coverter box. A VGA->HDMI converter box will probably run you ~$100 to ~$200 depending on your resolution needs, but will be your best bet for picture quality.

A few answers have mentioned S-Video, which you want to avoid. S-Video is very limited in the resolution it will carry, something around 480 lines.

There used to be a number of such adapters. The most popular was made by a company "Dazzle". Last I knew they were still around although you seldom see there products advertise or on store shelves any more. I had used a couple of these adapter maybe 7 or 8 years ago. They were very good for their time. Their products would work when you have an older PC or older TV without a more modern interface.

Most laptops come with an s-Video out port on the back. Check to see if you have that.