You can't explore Shakespeare's world without a visit to Stratford - but to be honest it's so overrun with tourists it's hard to recommend. The Birthplace Museum and RSC Theatre are worth a visit.
Unfortunately the old schoolroom in the Edward VI Grammar School isn't open to the public.
The replica of the Globe in Southwark, London, is well worth a visit. It's very atmospheric, and the guided tours are informative.
It's well placed for other attractions; Tate Bankside, the Clink, the replica of Drake's Golden Hinde, and the Old Operating Theatre are all nearby.