Click the "Purchase" link in the top menu of The Bluebook's website to buy the book in print or to buy a subscription or renewal to the Bluebook Online.
From the website:
We are no longer able to sell the 19th edition (ISBN 978-0-615-36116-1) through our website. If you wish to purchase the 19th edition, send an email to firstname.lastname@example.org. Please indicate if you are a bookstore, how many copies of the book you wish to purchase, and the billing and shipping addresses. You may also fax your order with your credit card number and expiration date to 617-495-2748. Make sure you indicate clearly whether you wish to purchase the 19th or 20th edition on the fax. We do not accept orders over the phone. Sorry for any inconvenience.
If you purchased a subscription to the Bluebook Online before the release of the 20th edition, you will have access to the 20th edition without any additional fees, and you will still be able to access the 19th edition as well.
For a concise compilation of changes from the 19th to the 20th edition, see this document from Cynthia Pittson at Pace Law Library. The biggest change, in my opinion, is that available at is not longer used for parallel citations to Internet sources. All Internet citations are treated as direct. It also encourages the use archival Internet sources such as perma.cc or web.archive.org.
And while we're on the topic of the Bluebook, you may want to regularly check the Bluebook updates provided on the website. The updates provide corrections to the rules and add clarification.