The Buckinghamshire County Museum is in the town of Aylesbury and has good access by train and bus. Parking is a bit more problematic but there are plenty of multi-storey car parks in the town and there is free parking at Aldi for 2 hours.
This museum has a great courtyard and is experimenting with Summer Sunday opening. The Roald Dahl gallery is a good alternative to the pricey museum in Great Missenden and there is a free exhibition on Islamic Art at present which is really good. Plenty of room for picnics so you can avoid the newly opened tea room if you want a no spend day !