6 Nights, Cambridge to York by public transport

One of my favourite areas of the whole UK is the north Norfolk coast- I suggest having a look at that. There are lots of nice small villages on either side of Wells-next-the-Sea, and Blakeney and Cley are lovely.

Castleton has pretty good bus services- hourly to Chatsworth and Blue John, also regular services to Bakewell, Edale Station and Sheffield.
This year red routes (don't even think about stopping) have been introduced on the Man Tor roads, plus parking charges and more double yellow lines in Castleton as the County Council's patience has snapped with the car dependent culture.
So it is, and has been for many years, possible to rely on public transport there.

Ely is so close to Cambridge that the one can substitute for the other.

Kings Lynn would make a good base for North Norfolk- well under an hour from Cambridge by hourly train, and a direct bus to Peterborough for York hourly. That can be, and is cheaper, booked as a through ticket from Kings Lynn bus to wherever.

Ely is 14 minutes by train from Cambridge, hardly worth moving base if you are in Cambridge. Even Sandringham is easy as a day trip from Cambridge. Norwich is probably stretching it slightly - but what about Kings Lynn, once England's biggest port and part run by the Hanseatic League, who left their old buildings. Be warned, Ely is notoriously difficult/impossible for luggage which can't be taken on the cathedral tour.

Peak District could be a little out of place in this scenario - but not impossible. Rail routes are limited to a commuter line from Manchester to Buxton, which is the wrong direction for Cambridge, and a line passing through the Hope Valley connecting Manchester with Sheffield, which is slightly easier but most of the stations aren't quite in the village they serve. There are also local buses from Derby via Bakewell to Buxton basically following the A6 road. Castleton has a local bus from Sheffield and there are various small buses (more like large hotel shuttle vans) squeezing their ways down various routes in between all the bigger villages. Cambridge to Sheffield can be done with just one change of trains but take around 3 hours, Derby is half an hour less. To this you need time to find the onwards buses and then maybe an hour of travel. I'd say aim for somewhere on the Castleton bus route because you need to be back in Sheffield to move to York - or Buxton and onwards by train to Manchester for connection to York. An American's view of visiting Edale in the Hope Valley (he started from Manchester) https://www.youtube.com/watch?v=9ud9-geNDvs