What common sayings and idioms did Shakespeare invent?
There are - literally - hundreds, even including some of Shakespeare's titles ("As You Like It", "All's Well That Ends Well"). Shakespeare made a major contribution to teh wordstock and idiom of English.
I've given you a handful below, and there are more in the - excellent - Bernard Levin quote at the link below.
Hope it helps!
It's Greek to me
more sinned against than sinning
vanished into thin air,
budge an inch
playing fast and loose
making a virtue of necessity,
laughing (oneself) to stitches
having too much of a good thing
...and a few more...
The Dogs of War
Beware the Ides of March
Something Wicked This Way Comes