As it happens, Great Missenden has a direct train service (on Chiltern Railways) to London Marylebone, and it's easy to reach Stratford upon Avon! However, the rest of your route could be tricky without a hire car. One-way hires tend not to happen so much in the UK, so they're expensive, but you may want to look at Sixt, who are more reasonable with the one-way surcharges (and have their site handily located next to Hatton Cross).

Personally, I find Stratford (unless you're specifically there to see a play) and Stonehenge very disappointing - in the West of England, I'd be more inclined to head to the Devon or Cornish coast.