There is no need to copyright your book (with the U.S. Copyright Office) before submitting it. Always remember that the moment your work is in tangible form—once you type it, save it, write it, or otherwise commit words to paper—it is protected under copyright law without any formal registration, even if you do not use the copyright symbol. However, registering your work with the Copyright Office (which does cost money) allows you greater power to litigate and collect damages should someone steal your work.
Any publisher who contracts with you will copyright the book in your name. A clause in most contracts between publishers and authors sets up an agreement whereby the publisher takes out the copyright in the name of the author. The publisher merely handles the paperwork on behalf of the author, and the copyright is the author’s property. (The author’s name follows the copyright symbol on the copyright page.)