register it with the WGA AND copyright it with the U.S. Copyright Office.
Yes, you need to do both because they protect you in different ways.
Registering your material with the WGA provides dated proof your ideas exist in the form of your script. It establishes evidence. If someone steals your idea, the WGA can't do anything legally to help you.
That's where copyrighting comes in. If you copyright your script with the U.S. Copyright Office and someone steals your idea, you can claim copyright infringement. Lawyers can step in and do their thing. Whether or not suing for copyright infringement is a successful tactic is a subject for another day...
(I've been told I'm protected if I mail myself a hard copy of my script through the US Postal Service, but that feels a little too DIY for my taste. I prefer a registration number.)
I consider the moment when I register and copyright my script as a celebration of its completion, so I lay down the money happily like I'm buying my script a drink.
For more information, check out the WGA Registry FAQ and the U.S. Copyright Office FAQ.
And check out these blog posts that also discuss the topic --