If you write a screenplay, go ahead and register it. You'll probably have to before a legitimate agent or producer will read it, anyway. And then you'll have to sign a release on top of registration.
But for a novel, registering the copyright is nothing but a waste of time and money. No one is going to steal your novel. It just doesn't happen. You already have the work copyrighted the moment you write it, and there's no reason at all to register it, while there are reasons not to. It is just a waste of time and money.
And, of course, if you do register the work, you are then forced by law to place the registration notice and date on the manuscript whenever you send it out, or the registration doesn't protect you, anyway. This alone can cause problems, especially if it takes a couple of years to sell the novel.
And many editors and agents just do not want to see the copyright symbol and registration date on a manuscript, anyway. It smack of amateurism and a lack of trust. Plus a lack of knowledge of how the business works.
The simple truth is no one is going to steal your manuscript because it holds zero value, and even if someone did steal it, there's nothing they could do with it. Unlike a screenplay, a novel manuscript is absolutely worthless until and unless a publisher buys it, and even then it probably won't earn enough money to matter to anyone, and the person will then be certain of being caught.
And when your novel is published, the publisher registers it for you. So unless you plan on self-publishing the novel, forget all about registration. It does you no good at all, and can do you some harm.
And plot ideas can't be stolen. You can't copyright either plot or ideas. There are thusands of novels and screenplays with the same plot and the same ideas. Unless the person uses your words and/or your exact images in expressing the plot or idea, it isn't a crime.
But this is neither here nor there. Novels and screenplays are two very different things. SCreenplays should be registered (And even with screenplays, it's usually more for the protection of the agent/producer than for protection of the writer), but novels not only shouldn't be registered, doing so can cause you some harm.