Write valuable content that will make people want to follow you. Make sure to have a like button on your blog and also a link to your facebook fan page. The key to attract and keep new visitors all comes back to the content you write.
You could also do paid marketing on facebook but if your not pushing a product or service then I'm not sure how you'd be able to justify that.
Once you have facebook fans don't just spam them with your brand. Interact with them. Ask your followers questions or post things people want to read. Before you post anything, ask yourself if it's something you would read or respond to. You not only want to get more likes but you also want to retain them once you have them.