You can take a train from Bury St Edmunds to Stratford via Ipswich in around 1 h 59 min. Alternatively, National Express operates a bus from Bury St Edmunds to Stratford once daily. Tickets cost $11 - $15 and the journey takes 2 h 40 min.
From Bury St Edmunds train to Ipswich then train to Stratford (London). 1 h 59 min £36. From Bury St Edmunds bus to London Stratford. 2 h 40 min £10. From Bury St Edmunds drive to Stratford. 1 h 17 min £13.