Song royalties are paid to both the composer and the publisher of the song. If you expect (or hope) to get any air-play, or to have your songs covered by other artists then you will probably want to form your own publishing company. You will also want to become a member of either ASCAP or BMI, which are entities that police and collect royalties on behalf of composers and publishers. If you contact them they probably can provide you with a kit that will tell you all that you need to know. I believe that you must have a publisher in order to register your songs with ASCAP or BMI.Of course, you will also need to copyright your songs. The federal copyright office will probably send you all you need in order to do this.
Setting up a corporation is neither very expensive nor very difficult. Most attornies will do the whole thing, including the fees that must be paid to the state, for $1,000 or less, or you could probably do it yourself with information from the Secretary of State of other appropriate government office in your state. Your local reference librarian would also probably be able to guide you to a book with forms for your state. Corporate status also requires various annual filings, including for taxes, since you will have created an artificial person.
It may also be possible for you to set up your publisher without incorporating, i.e., as a sole proprietorship which you own and are responsible for. You should check with your accountant to see whether this would require you to obtain a separate tax number (different from your own social security number) and to file a separate tax return for the enterprise. It may only require you to file an additional schedule with your personal income tax returns.
I am a New Jersey attorney and I do not practice in this area, but I believe the above describes the basics. If you are really serious about the songwriting business then you should probably invest in a full consultation with a lawyer or other professional so that you can be sure that you understand the way things work. You may also find it worth your while to hook up with an established publisher.
Good luck.
Bennet D. Zurofsky, Esq.