Should I copyright music that I put on the web or online?

If you are concerned about people stealing it or using it without permission, then yes.

Under U.S. law, an original work is protected from the moment it is fixed in a permanent medium. Either the words or the lyrics or both together can be copyrighted this way. In addition, your performance of the music can also be copyrighted. Although it is not required by law, you can register your copyrights with the U.S. Copyright Office (Library of Congress) which gives the public further notice that you own this material and impacts the amount of damages owed to you by violators. So you can post your music to on-line sites, knowing that the material is protected already. However, once posted, it is almost impossible to prevent someone from violating your copyright and very difficult and expensive to recover damages. So don't post anything that you think is a real money-maker.

Once you create the music and put into a tangible medium - you own the copyright. Registration of your copyright is only required if you want to enforce your rights in Federal Court.