There are no qualifications for being a professional magician. Some magicians take acting classes to help with their onstage performance. Taking a couple of business classes wouldn't be a bad idea for helping you navigate your way through the entertainment industry. Furthermore, some schools or camps, such as the Academy of Magical Arts, offer basic magic classes. You can also try to enroll at Hogwarts, but muggles need not apply.
Oftentimes, people get into magic by shopping at their local magic store, learning tricks from books and practicing in front of people. People rarely apprentice under accomplished magicians. However, it is not a bad idea if you get a chance. Magician organizations such as The Society of American Magicians offer magicians—non-experienced and experienced alike—a community of avid magic fans.