Often, the best way to keep up with a favorite author's publications is to go directly to the official website or social media accounts. On fantasy author Kalayna Price's website, for example, she has a blog, an email newsletter for fans, and links to her social media accounts.
The author's publisher's website is another good place to check, as some offer subscription services, which send out notices when a new book or story is published. Harper Collins is one example, with its Author Tracking service.
Goodreads is another popular site through which fans can track an author's activities. It is a social media site dedicated to books and authors. One does not need an account with the site to be able to search for authors and see their posts or fan groups' posts about the authors' work. Authors typically have a special account on the site, which allows them another venue through which to update fans on new books that are coming out. These sites are viewable by anyone, whether a member or not. Those who choose to sign up with Goodreads can then write book reviews and mark items on a list of future reads, along with many other features.