The simple answer to your question "How can I get a career in art" is you make one. All arts are intertwined. A song is a three minute play (musical). Decorating is an art, the combination of architecture and design within a three dimensional space. Form, shape, colors and light used together to make a statement. This is the same in theater and film settings as well as homes and businesses.
How you chose to go is up to you. If you want to paint and only paint in one style then find what people will buy and stick with it. There is a downside to that way though, once they start to buy, you will not be allowed to grow beyond that style. But if you are still looking for your type of expression I would suggest you try everything and anything. Absorb as much as you can. All you do will be useful someway or another. It might surprise you to know that few artists make it doing what they first aspired to do. How they ended up making it in the arts was simple, they were there when opportunity happened by and they said sure why not?
That's a great question. There are many ways to get a career in the arts but there are also many different types of art. It would be good to know what kind of art specifically that you're interested in to better answer your question.
For example, there are visual arts, performing arts, literary arts just to name a few. For the sake of this question I'll assume that you're talking about visual arts but some of this advice applies across the board.
First off you must practice. In anything in life and of course in art you need to continually practice and strive to get better at your craft. Take your classes seriously. Use critiques and constructive criticism to improve and get a better understanding of color, composition, design, whatever it is that you need to work on.
If at all possible, get an internship at a company in the field that you want to pursue. Or find a mentor who can give you guidance and feedback. Seek out peers and friends who will give you honest feedback and not just tell you what you want to hear.
The most important thing to landing a career or job in the arts, however, is having a great portfolio. That will be the way prospective employers get exposed to your art. Make certain that you only include the best of your work. But also make sure to include process pieces. Show how you came to develop the piece(s) so that if you have to work at a company or with a team, they'll know what your process is like and if that fits with their job opportunity.
Finally, it doesn't hurt to be good to work with. You might be the best artist out there but if people can't work with you then they won't hire you at a company. You can get away with that kind of stuff a little easier if you're a fine artist.
I hope that some of this information helps. Good luck out there.