One always needs to be careful when talking about "mistakes" in a particular edition of Shakespeare. There are many varying "original" editions of Shakespeare's plays, and it is only in a few cases that one can claim that a particular edition is more "authentic" than another. It is also important to realise that in most Elizabethan texts the punctuation was decided by the printer more often than by the author; punctuation varies significantly from copy to copy and it is likely that manuscripts and prompt books had little or no punctuation.
The source of these texts is the Mody (tm) Shakespeare. It is provided by many people on the net, including Project Gutenberg.
As an exercise, let us look through the text of Hamlet's "To be or not to be" speech compared to the Arden Hamlet. The differences are highlight below, but if you're interested in this yourself you might get a good scholarly edition of Hamlet which lists all the differences between early texts; some good choices are: an Arden single-play edition, the complete works edited by David Bevington, or the Riverside complete works.
So this is where the speech on the server differs from the speech as edited by Harold Jenkins for the Arden edition. The text is the one from the server, a line beginning with > indicates a difference.
To be, or not to be: that is the question:> a period instead of a question mark
Whether 'tis nobler in the mind to suffer
The slings and arrows of outrageous fortune,
Or to take arms against a sea of troubles,
And by opposing end them? To die: to sleep;
No more; and by a sleep to say we end> a colon instead of a comma
The heart-ache and the thousand natural shocks
That flesh is heir to, 'tis a consummation
Devoutly to be wish'd. To die, to sleep;> the first colon is a comma, the second a dash
To sleep: perchance to dream: ay, there's the rub;
For in that sleep of death what dreams may come> a dash instead of a colon
When we have shuffled off this mortal coil,
Must give us pause: there's the respect
That makes calamity of so long life;> Jenkins chooses "dispriz'd" instead of "despised"; "dispriz'd" comes from the Folio, "despised" from the second Quarto
For who would bear the whips and scorns of time,
The oppressor's wrong, the proud man's contumely,
The pangs of despised love, the law's delay,
The insolence of office and the spurns> Jenkins chooses "pitch" from the second Quarto; "pith" is from the Folio
That patient merit of the unworthy takes,
When he himself might his quietus make
With a bare bodkin? who would fardels bear,
To grunt and sweat under a weary life,
But that the dread of something after death,
The undiscover'd country from whose bourn
No traveller returns, puzzles the will
And makes us rather bear those ills we have
Than fly to others that we know not of?
Thus conscience does make cowards of us all;
And thus the native hue of resolution
Is sicklied o'er with the pale cast of thought,
And enterprises of great pith and moment
With this regard their currents turn awry,
And lose the name of action.--Soft you now!
The fair Ophelia! Nymph, in thy orisons
Be all my sins remember'd.