But to answer your question the RAF Museum at Hendon is only 20 minutes from Luton on the Thameslink rail line (free shuttlebus to the station from LTN airport) but I think you'll struggle to get to Duxford by public transport, the UK is not that well organized for non car owners...Hire one if you can.
Both sites are superb, the RAF museum is all static with some fantastic rarities and Duxford is much larger with lots of flying aircraft on view. To get really up close go to the Shuttleworth Collection near Bedford where there is also a Spit (albeit with very odd shaped wingtips...), plus a couple of score of the rarest pre WW2 and several pre WW1 aircraft on display, most flyable.