Check the copyright page first. It's the verso (left) page after the title page, and includes, along with publisher and city, both the copyright and publication dates. These are often interchangeable, though they sometimes differ because of reasons like printing delays or sales decisions.
Look for the phrase "originally published" accompanying the date if you're looking for the first publication of the material. Clarify that it isn't just the copyright or reprint/translation date. If you're citing the book as a reference, take the latest publication date found on the copyright page, since it's the version you used as a source.
Peruse the first or second title page of books printed early in the 19th century, which might include, move or omit the publication date altogether. The second page contains more information, so you're more likely to find the publication date there. If not, check the book from front to back. For a loose guess, you can use the date -- if included -- with the author's name after a foreword or introduction. You might also see a publication date imprinted on the front or back cover. Remember, you're looking for publication, not copyright.
Go to the Library of Congress online catalog if you can't find the date in the book itself. The catalog clearly specifies both the copyright and publication date for most documents. For example, in the print version of "Crusade in Europe" by Dwight D. Eisenhower, the year 1948 appears next to the publisher on the title page. But the Library of Congress catalog specifies 1948 as the copyright date and 1952 as the publication date. This year is not listed anywhere in the print version. You can research books by title, author or ISBN.
Research the publishing company online, keeping in mind that the name may have changed. If the publisher is still in operation, you might be able to find an online catalog of previously published books.
Find an autobiography of the author or research the author online. Materials about authors' lives often include a chronological list of published works.