Good news: I have a list for you.
Of course this list doesn't help you a bit if your favorite authors are dead, but we are supposing you have a modern author you admire. This, then, is the skinny:
#1: Buy their books. Look, we are flattered when you borrow one of our books from a friend and love it, but it does very little for sales. What really helps is you buying our books for yourself, putting them on your birthday and Christmas lists, and buying them for friends who are having birthdays or Christmases. Buy. Our. Books. It's super helpful. It makes them go up in the ranking on Amazon. It gets them on the New York Times Best-Seller list. It's helpful.
#2: Review their books intelligently. When someone is waffling about Position No. One, they might click on the reviews tab for that book on Amazon, or look it up on Goodreads. I know I do. And when I look up a review, I don't want to see all caps-lock, "OH MY GOSH IT WAS THEBESTTHINGEVERRRRRRR." I am looking for someone who actually seemed to have their wits about them while they were reading, who can tell me something (but no spoilers) about the plot, the quality of the writing, and what I liked best. I love hearing what other people liked best...and if it happens to be something I'm inclined to like best, I'll probably end up buying the book.
#3: Start a blaze. Carefully. No, please don't burn our books. But if you tell your friends enough about the book (and buy a copy for them, maybe) then they'll read the book, and if they read the book and enjoy it, they'll tell their friends, and pretty soon you'll have started a wildfire which is extremely helpful for your favorite authors who are, in this way, fans of pyromaniacs.
#4: Give them a hand on Twitter, Facebook, and other social media. I can't tell you how much I love a few of my followers on Twitter who, no matter what sorts of links I share, will share them from me. And not just retweets, but an actual, "I went to the site and tweeted it myself" sort of thing, while still mentioning my name. Those people are valuable because they care enough about what you said today in a blog post, or what you recommend they read next, or whatever it is. They care enough about it to share it in their own words and ways. It's precious. (Also, liking a post on Facebook brings it up in other peoples' feed. That's actually how I've found several of my favorite pages to follow. It works on authors' feeds too! ;)
#5: Surprise them. Several times since my debut novel, readers have surprised me with an email in my inbox, writing about what they enjoyed in my book, or how they came to read it, or something else interesting. One reader emailed me an amazing drawing she'd done of the main character (I happen to already be a fan of this girl's art. It was good). Sometimes a reader will post on the page, or message you via Facebook or Twitter, or something else. It's uplifting to the author to get a message like that and not expect it. We don't need the reader to gush, but it is heartening to hear that, independent of your reach, a new reader got hold of your book and enjoyed it. We thrive on approval ratings.