Many writers who want to focus on creative writing find themselves getting discouraged. The truth of the matter is there are more markets for non-fiction, copy, and technical writing etc. However, with creative thought you can come up with many ideas for selling creative content.
There are markets out there and even contests for short stories, poetry, and even movie scripts. I recommend getting the latest copy of the Writer's Market guide to discover the better paying markets for creative writing.
Creative writers often have to develop the most patience and persistence. They also tend to not be paid as well as non-fiction writers due to supply and demand issues. Creative writers are those who truly enjoy the passion of creating and story telling, unfortunately not everyone is going to be the next rags to riches J K Rowling. My goal of course is never to discourage anyone, but with creative writers there is typically a lot of rejections and failures before success comes. To make it as a creative writer you have to really have the fire and passion to do what it takes even when the road is bumpy (or non-existent!)
Are you a lyricist or song writer? sell song lyrics. Write poetry or other meaningful writing and sell it to greeting card companies. Write a novel and go the traditional route with a literary agent. Although most non-fiction writers can work successfully without a middle man, if you want to publish fiction, chances are you are going to need someone with connections to even get your manuscript read. The slush pile is not the place your hard work should be buried, so a good agent can be a worthwhile investment.
Perhaps get your feet wet by writing and illustrating a children's story and self-publishing. Volunteer to do readings at the local library or schools and market your book that way. Take creative writing and marketing courses. Many creatives don't like to focus on the nitty gritty marketing stuff, however that knowledge can be what makes or breaks your creative writing career.
Self-publishing often gets a bad stigma, but what better way to show off your talent than to have something in print that says "look what I can do?". I believe self-publishing or even an ebook can be a great way to market your talents. Also, consider doing something free online like "hubpages". Hubs are a great way to share a chapter of a story or some poetry. People can read your work there and share it with others which can lead to better things down the line. If an agent or publisher sees a lot of people following your work on a website like this, it can encourage them to give your work a shot over others. The other upside to a website like hubpages is the support of other writers.
Another idea is slogan writing or jingles that you can pitch directly to companies. Although this could be considered a form of copywriting as well - it also has a creative bent to it that is often suited to creative writers. I heard it on good authority that many catchy slogans have garnered their writers up to a million dollars. Think of some of the better ones - when you here "no one can eat just one" for example I bet you immediately thought "Lays" brand potato chips. "Finger Lickin Good" - KFC, the list goes on. If you can write killer slogans you can earn really good money.