If you want to publish it for real, don't post it chapter by chapter on Facebook. Keep it for yourself until you're ready in whole. If you just want to write and share you're writing, then go ahead.
But you have different ways to go about publishing for real, depending on what you want out of it.
There is self-publishing - I agree with SAColten - lulu is a great site. (You pay the publishing fees - and this is mostly to have the book in hand, and sell it or give it in smaller quantities. There are exceptions to that.)
There is e-publishing. Some sites are very particular, some sites don't care as much about the quality. (Got to research) (This will get you published around the internet or on booksellers like amazon. You can definitely get a fan base here, and some extra cash)
There is traditional publishing - these are the books in the bookstores. (fan base and cash here, but it takes longer)
First thing you want to do is write the book.
When you've written and are ready to publish, make sure your genre is identified. Horror, sci-fi, romance, mystery, or contemporary fiction, etc.
Then you find the publishing places that publish your genre of book.
Its a lengthy process to get started, but well worth it.
Also - you can self publish e-books with amazon, and upload them directly onto their Kindle (e-book reading device). That's getting popular.