Quite honestly, it depends on the criteria you employ. This informative piece from the Sturgis Library shows how different institutions can claim that title when using different “first” questions. Most consider the Library Company of Philadelphia to be the first subscription/lending library. Founded in 1731 by Benjamin Franklin, it still exists as a vital/valuable tool to those who work in Americana; it has extensive collections in 17th to 19th century American texts. You can peruse its Bulletins to see how extensive and active its acquisitions program was with both monographs and serials; a reprint of its 1741 catalog offers additional insight into the first years of collection development; and this 1807 catalog includes the charter, laws, rules and regulations of the library. In addition, this catalog lists all the monographs in alphabetical order by author, but the books are arranged on the shelves by size: folio, quarto, octavo, and duodecimo!
What Was America’s First Library?