The British Library has numerous resources for a search on William Shakespeare; the results are listed in four categories (web pages, online gallery images, catalogue records, and journal article).
The Furness Shakespeare Library at the University of Pennsylvania Libraries maintains English Renaissance In Context (ERIC) which has two parts: ERIC Introduction & Overview and ERIC Tutorials. Some of the tutorials focus on major textual issues concerning Shakespeare's plays.
The Shakespeare Birthplace Trust has the world’s largest publicly accessible library and archive devoted to Shakespeare, the sources of his plays and their history on the stage. They also provide access to photos, reviews, and DVD recordings from the Royal Shakespeare Company.
The Folger Shakespeare Library sponsors the Folger Institute; this part of the Folger Shakespeare Library's website contains information about the Folger Institute, program offerings, and application guidelines and deadlines.
The Huntington Library's collection includes the First Folio published in 1623; this research library is a private, nonprofit institution. In order to use the library's collection (reading privileges) you will need to register as a reader.
The Newbury Library is a non-circulating library with over 1,400 materials by William Shakespeare in its collection; the library focuses on researchers who visit in order to use the library's extensive collection.
The Shakespeare Centre Library and Archive (part of Shakespeare Birthplace Trust) in Stratford-upon-Avon, UK contains a vast amount of material concerning Shakespeare's life, his works, and performances of his plays.