Qualifications
There are no qualifications for being a painter. However, it helps if you go to college and bone up on art history or painting. It's important to know how to paint and why people paint the way they do.
Some people continue on to get an MFA. Why get an MFA? For one, you'll be able to teach at a university. Most colleges, art schools, and universities only hire professors with an MFA. It's also easier to get a job in a museum or art gallery with an MFA.
People also go to graduate school to make important connections. If you thought high school was over, think again. The art world can be like a big high school. You want to sit with the cool kids, because the "losers" aren't even allowed a mile near the schoolyard. Your classmates might be the people who will help you get a job in an art gallery or introduce you to collectors and gallerists.
Furthermore, people get their MFAs so that they can create a body of work. You get two to three years to create your first big body of work without dealing with the real world. A lot of people get student loans and camp out at their studio for years. You'll get a real taste of what it's like to devote most of your time to art. You might actually find that you hate it. That bit of knowledge might save you from years of struggling to produce artwork.
The last big reason to get an MFA is to work with a professor or mentor (source). These people will tell you if they think painting your cat is lame. You'll get criticism that's worth its weight in gold if you go to the right MFA program. So how do you choose an MFA program? You go to one that has professors you admire and a program that will help facilitate your creativity.