Understand that a paperback is rarely going to be considered a "First/first," which is the term used by collectors to refer to the book's first edition, first printing. Unless there was no hardcover for the book in question, a paperback will never be considered by a true collector as a first/first, even if it is the first paperback printing.
Check the publishing information as you would for a hardcover. Look for a stamp designating the book as a first edition. Even though the paperback follows the hardcover by as much as a couple years, if the typeset has not changed--that is, if the content is a faithful reprint of the original without changes, additional forwards or anything else--then it is still the first edition and most publishers will stamp that information on the page.
Identify the printing number by checking the number string below the publishing information. As in the hardcover, the paperback includes a line of numbers one through nine or one through nine plus a zero. After the first printing, the one is removed. After the second printing, the two is removed. The lowest number to begin the string is the printing number.
Know your terms. There are trade paperbacks, mass-market paperbacks and book club editions (BCEs).
Trade paperbacks are usually the same size and shape as a hardcover, and they're sold by book retailers only, though that is changing. They will usually carry the first edition paperback stamp.
Mass market paperbacks are the easily held variety found on bookshelves in stores and seen folded over backwards by sunbathers the world over. These can contain all the original text, but will rarely be considered first editions by collectors.
BCEs are often condensed, and they are designated as nothing more than BCEs regardless of its publication history.
Expect confusing variations in the edition and printing notations. For example, Random House prints the words "First Edition" above the number string, but the lowest number is two. When they run a second print, they remove the first edition stamp, and the lowest number on the string is still two. It's still a "first edition, second printing," but without the words "First Edition," you'd be hard pressed to tell.
Always bring your copy of Bill McBride's seminal resource, "A Pocket Guide to the Identification of First Editions," when you do your paperback book shopping, especially if edition and print numbers are important to you.