How well was Shakespeare's work received in Elizabethan England? Was his work groundbreaking and better than his contemporaries, or was it largely the same as other works of the time?

There's a lot more to be said on this topic, but you might enjoy these previous threads:

My usual answer to this is basically that Shakespeare's work was well-received but he wasn't the only well-received playwright of his day -- there's few of the formal rubrics modern entertainers can draw on to demonstrate their critical reception, like box office statistics or film reviews, but sources like references to Shakespeare in contemporary writings and financial records of playing companies putting on Shakespeare's plays situate him alongside other prominent playwrights not as the undisputed best but as one of several options for the Elizabethan and Jacobean reader and viewer. His early narrative poems Venus and Adonis and The Rape Of Lucrece were wildly popular, going through many editions and reprints in the decades after their publication, and seem to have made a broader impression on the public imagination than any single play -- for an interesting look at what this could look like, you can read about some unusual readers and listeners in Katherine Duncan-Jones' ""Much Ado With Red And White: The Earliest Readers Of Shakespeare's Venus And Adonis (1593)".