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, 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 a fee -- usually require payment to a performance rights society such as the American Society of Composers, Authors and Publishers (ASCAP), or Broadcast Music, Inc. (BMI).
If you'll be playing the music in a venue such as a restaurant or tavern, you don't 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. Purchasing the sheet music also doesn't give you the right to record the song and distribute copies of it or to copy the sheet music.