There are no actual qualifications to become a professional magician. It really does come down to your stage presence and ability to perform the stunts.
Some magicians take acting classes to help with their onstage performance, and business classes wouldn't be a bad idea to help navigate the dangers of entertainment contracts and merchandising agreements. There are also magician schools and camps, such as the Academy of Magical Arts, which offer basic magic classes to the aspiring masters of illusion.
You could try enrolling at Hogwarts, but unless you have something more than a sweet handkerchief routine, you might want to forego that application fee. Muggles need not apply.
Oftentimes people get into magic by shopping at their local magic store (yes, they exist), learning tricks from books, and practicing in front of their family and less-judgmental friends. You may think that getting an apprenticeship is a logical idea, but that's not so common.
These days people rarely apprentice under accomplished magicians. There are such things as magician guilds however, like the Society of American Magicians. But just because it sounds like something fantastic, it doesn't mean you're going to be anywhere near the Seven Kingdoms.