You automatically own the copyright to any song you write but if you really want that to mean something worldwide you should register it with the U.S. Library of Congress. It's about 10 bucks a song or something.
Socan generally won't register a song unless it's performed publicly (TV, radio, CD, live performance) in a way that would recieve royalties since that's what they do, distribute royalties for composers and publishers.
As for publishing, you can self-publish and start a publishing company for a small registration fee. If, for example, you release an indy CD and you own your own publishing it puts you in a better bargaining position if you happen to land a bigger deal with a record company who will want a bigger chunk if they are previously unpublished. Hang on to your publishing if you can, look what happened to the Beatles. Michael Jackson owns it!
