All you need to get a library card is a photo ID showing your current, local address. If your ID doesn't have your local address on it, you'll need to bring something that does.
In addition to the fine collection of print fiction, you'll also be able to digitally borrow thousands of eBooks and audiobooks.
Bookstores and most public libraries shelve fiction by genre and the last name of the author. If you wanted to find a Stephen King book, for example, you'd probably head for the Horror section, then stroll along to the Ks.
Academic libraries use a different system: Library of Congress Classification (LCC). Browsing under LCC requires a different approach. Books are shelved by subject with call numbers that begin with a letter or two and a string of numbers, and there's no special section just for fiction. Your hypothetical Stephen King book might have a call number of PS3561.I483 T66 1988—which, you might notice, doesn't even have a "K" in it!
Generally speaking, English-language fiction is found under the PR and PS sections. PR contains English (as in from England) fiction, and PS contains American fiction. Fiction originally written in other languages is shelved in a section for the original language, regardless of whether it's been translated. Books by Gabriel García Márquez, for instance, are shelved in the PQ section. The full breakdown can be found in this classification outline.
Within these broad categories, you'll find subsections for different forms of literature (such as poetry and drama) as well as for different time periods. To further confuse you, comic books and graphic novels can be found in the PN6700s, though the rest of the PN section largely consists of works about literature.
Given the complexity of physically browsing the stacks for fiction, you'll find it much easier to do your "browsing" online in the catalog. And if you're looking for a particular author or title, the catalog should always be your first stop; otherwise, you'll just waste time trying to guess where the book you want is on the shelves.
To learn how to do this, click on the tab for Searching the Catalog.