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Shakespeare being such a prolific author must have left behind a tremendous library of books, letters, rough drafts of plays, scribbled notes and writings of every description.

Having read The Millionaire and the Bard, I can answer the question. First, you must understand that in the late 1500s and through Shakespeare’s 1613 death, paper - then made from cloth, not wood pulp - was expensive and scarce. Secondly, in Shakespeare’s England there was no such thing as copyright.
Consequently there was a distinct tendency to reuse paper, on the one hand, and a distinct incentive not to publish the script of your play, on the other. Publishing your play would simply enable competing play producers to plagiarize, and you would have no legal recourse against it. Consequently none of Shakespeare’s plays, and precious little other material, were published during his lifetime. In fact, we would not now have any of his plays - and we know that we do not have them all - but for the efforts of his friends/comrades (Shakespeare was an actor as well as a playwright) to publish them. Well, that is an exaggeration; someone else not connected with Shakespeare published some of his plays, tho not all in the cannon, and not word-for-word the same as in what is known as the “first folio.” (“Folio” refers to the format in which it was printed, which was the simplest but not the cheapest way in which it could have been done).
This “first folio” was created as in effect a vanity project; his friends were not actually trying to make money but had to try to break even because it was an expensive project. They decided to go for the top end - their only hope of breaking even was to position the resulting book as a mark of serious couth (as in, the opposite of uncouth). Only 750 first folios were printed, and they were printed on expensive paper. With the result that even the first one printed, which would in modern parlance be kept perhaps as a keepsake but would be marked up and used to correct typos and such, was sold.
As might be expected, some serious attrition in that 750 number has occurred in the past 390 years or so since the folios were printed starting about ten years after his death. Relatively speaking, Shakespeare was a prophet without honor in his own country. Shakespeare came into cultural vogue in a big way in Nineteenth Century America, with the result that there came to be a public outcry in Britain over the sale of first folios to American millionaires. Said millionaires reacted by using agents to represent them, so their purchases were not public knowledge.
Of those American millionaires, one was a formerly impecunious student named Folger, who became John D. Rockefeller, Sr.’s right hand man and later chairman of Standard Oil. Folger (no relation to the coffee) became a Shakespeare fanatic as a student in Amherst College, married a woman who was like-minded, and as a couple established a relationship with an American professor of Shakespeare studies and picked his brain to learn as much as possible about Shakespeare artifacts and first folios. With that background and the resources he was raking in, the Folgers (secretly, as much as possible) basically raided England of Shakespeareiana. His wife meticulously cataloged the stuff, and they paid thousands of dollars annually just to warehouse it.
So the answer to your question,

Where is this collection at so the interested may visit?

is that

Shakespeare aficionados can only dream of the existence of the trove of materials which you assume exists, and

To the extent that a trove of Shakespeare material does exist, it is in the Folger Shakespeare Library in Washington, D.C.
Something like 225 first folios are known still to exist; the Folger has about 83 of them. None of them are for sale, but the last known purchase of a first folio was for something like $5M. Each first folio is considered unique, and to have independent research value. One of them was in a British public library - available to be borrowed! - for years.
The Folger is close to where the SCOTUS bldg was later erected; Folger secretly amassed ownership of the lots over a period of years. Just when he was about to start building, Congress moved to condemn the area for an extension of the Library of Congress. Mr. Folger wrote the Librarian of Congress giving a hint of what he had, and saying that he had spent a long time acquiring that land after considering many other sites, some abroad and none in Washington. So the government had the choice of allowing The Folger Shakespeare Library to be built (and endowed) as planned - or condemn the property and know that the greatest Shakespeare library in the world would not be in Washington, DC. The librarian of Congress - and Congress itself - took the hint.
Admission to the Folger - which includes a First Folio on display (at any given time, most are in climate-controlled vaults below) and includes a theater and artifacts of Shakespeare’s period, etc - is free. I wanna go. I’m wondering if I can take a grandchild or two . . .