How do I protect my idea for a reality TV show? Would filing a copyright or patent work?

A patent would not protect a television show. A trademark could protect the name of the show, but a copyright would be what you are actually seeking, though a copyright protects the expression of an idea, not the actual idea itself. Basically, you need to actually have copyrightable material first not just an abstract concept.

You cannot protect an idea. Sometimes, there are several reality shows that are practically identical going at the same time. If you write a treatment or a script, you can protect the actual writing with copyright.

You will want to file for copyright protection for your idea. You should also have a nondisclosure agreement signed by anyone you tell about the idea.