Goodreads reviews come from a community of readers who track and discuss books, so the way to get them is to be part of that community, not to treat it like another sales channel. A complete author profile, some ARC outreach through Goodreads-active readers, and patience will get you further than trying to force reviews the way you might on a retail site.
This is part of the Book Reviews 101.
Before asking for reviews, claim your Goodreads Author profile through the Goodreads Author Program. You'll typically need to verify you're a published author, for example with your book's ISBN or a link to its retail listing. Once approved, make sure your books are correctly listed, add an author photo and bio, and consider connecting a blog feed if you have one. A complete profile signals to readers that there's a real person behind the book, and it's where interested readers will land after reading a review.
Goodreads prohibits spam and self-promotional content that abuses its features, and it expects authors to interact as members of the community, not purely as marketers. Read Goodreads's Author Guidelines and Community Guidelines before posting review requests in groups, since heavy-handed self-promotion can get flagged.
Goodreads has also tightened its rating and review policies for pre-publication books, including asking advance readers to disclose how they got their copy (such as from the author, a publisher, or a Goodreads Giveaway). Keep that in mind if you're distributing ARCs to readers who'll post on Goodreads.
Goodreads Giveaways let you offer copies of your book to members who enter, and it's a built-in way to reach readers actively looking for something new. There is a fee to run one, so weigh the cost against how many entries (and eventual reviews) you're likely to get. Not everyone who wins a giveaway will post a review, but a giveaway does put your book in front of readers who chose to enter because they were interested in it.
Goodreads groups organized by genre are often open to discussing new releases within their own posting rules, similar to genre-specific communities elsewhere. Listopia, Goodreads's community-curated list feature, is another way to get discovered: find lists relevant to your genre (for example, "Best Self-Published Fantasy") and, where appropriate, ask readers who enjoyed your book to add it or vote for it. Don't add your own book to lists yourself in a way that looks like spam; let genuine reader interest drive it.
Not every reader can post a review on Amazon; some retailers require reviewers to meet a purchase threshold, and reviews can generally only go up on or after publication day. Goodreads has fewer restrictions, and many ARC readers who can't (or don't want to) post on Amazon will post there instead. When you invite ARC readers, mention Goodreads as an option alongside Amazon. See What Is an ARC?
Even though Goodreads doesn't sell books directly the way a retailer does, many readers check an author's Goodreads page before deciding to buy, and other services, including some promotion sites and retailers, take Goodreads ratings into account. A modest, honest collection of Goodreads reviews adds credibility that complements what you're building on Amazon.
Reach More Readers Who Review
Entrada's ARC review service can help put your book in front of readers, bloggers, and other reviewers, many of whom are active on Goodreads as well as Amazon.
Claim your Author profile, make sure your book is correctly listed, and encourage your own readers and ARC recipients to post there, not only on Amazon.
No, reviews themselves are free. Running a Goodreads Giveaway does have a fee, and it's one way, not the only way, to reach interested readers.
You can participate in genre groups and review-request threads within their rules, but mass, unsolicited messages to strangers are treated as spam.
Amazon has its own requirements for who can post a review, and reviews there generally only go up on or after publication day. Goodreads has fewer restrictions, so some readers post there even if they don't post on Amazon.