There are 2 Sumter County branch libraries locate in The Villages -- Wildwood (Pinellas Plaza) and Belvedere Dr. near Buffalo Ridge
here is the website
Library Services | Sumter County, FL - Official Website
Services are very comparable to any large county public library
Also access to many online resources.
Parts of The Villages are in Marion County where the nearest branch is in Belleview.
Parts of The Villages are in Lake County where the nearest branch is in Lady Lake.