I'm a recent college grad, interested in making some money off all those years of violin lessons. When I purchase sheet music, am I also buying the right to perform the music for free or for hire?
No. Merely purchasing sheet music does not grant you the right to publicly perform the song. You can, of course, perform it privately among friends and family. But public performances, whether for free or for money, usually require payment to a performance rights society such as the American Society of Composers, Authors, and Publishers (ASCAP), or Broadcast Music, Inc. (BMI). These rights organizations manage the rights of artists and composers.
If you'll be playing the music in a venue such as a restaurant or tavern, you typically do not need to worry about the fees, since the establishment will take care of paying the performing rights organization. If you'll be performing the music as part of a religious service in a house of worship (and not as a social performance), no fee is required.
Another important exception is works of music that are in the public domain. As a general rule, any composition written before 1922 is no longer protected by copyright, and therefore you can play most classical music publicly without needing to track down Beethoven or Mozart's heirs!
You might wonder why your rights are limited, even though you bought the sheets of music. Copyright gives owners certain exclusive rights, including the right to publicly perform and sell their work. Under the first sale doctrine, a buyer (like you) is permitted to use a specific copy of the work in any manner desired. But that doctrine does not give the right to record the song, distribute copies of the sheet music, or otherwise profit from music that was written by another artist.