I've not been to Luton Airport but I know that there's a regular Thameslink train service run by
First Capital Connect that will take you straight to/from the centre of London.
I expect you'll find there's plenty to occupy you in London for 4 days. But of course there are many possible day trips:
- Stonehenge and Hampton Court are both excellent suggestions.
The following are all easily reachable in one day:
- the university towns of Cambridge and Oxford,
- the cathedrals at Canterbury, Ely (near Cambridge), Salisbury (near Stonehenge), Winchester, Chichester
- the lively and characterful seaside town of Brighton
- many quaint country towns such as Lewes.
Generally it is quickest to travel by train. However, do be aware that trains in England are expensive!