Where to go?

Without knowing your interests it's not easy to advise but here are some possibilities:
York to 
Leicester, to see the grave of the recently-discovered King Richard lll (the 'king in the car park') in Leicester Cathedral (only made a cathedral in 1927...but the building largely dates from the 1000s), the nearby, excellent King Richard lll Visitor Centre which encloses the site of his grave:
https://kriii.com/
colourful, busy Leicester Market, the remains of Roman baths at Jewry Wall and to experience a massively, and comfortably, multi-cultural English city which has brilliant eating-places from a vast range of cuisines.
https://www.visitleicester.info/
b) Or Birmingham (England) with its thriving Canal Quarter (more canals than Venice!), its excellent museum and art gallery:
https://www.birminghammuseums.org.uk/bmag
the Jewellery Quarter with its fascinating museum...plus a multitude of shopping, eating and cultural activities.
https://visitbirmingham.com
c) or Stratford-upon-Avon Yes, you'll be amongst throngs of tourists but Stratford's an interesting and attractive town with much historical interest. It's certainly worth visiting ancient Holy Trinity Church even if you're not interested in seeing Shakespeare's grave:
https://www.visitstratforduponavon.co.uk/
d) or Northampton, which has several ancient churches (including Holy Sepulchre, an atmospheric Templar round church dating from around 1100) , the wonderful 78 Derngate, the only house in England designed by Charless Rennie Mackintosh, the lovely Guildhall and a multitude of shopping and eating options.
e) or Coventry which, although not a particularly attractive city, does have the ruins of its ancient cathedral plus its magnificent modern cathedral:
http://www.coventrycathedral.org.uk/wpsite/
a variety of ancient buildings which have been moved to and rebuilt in Spon Street, the Herbert art gallery & museum and the transport museum. 
There are other possibilities too (e.g. Leeds, Sheffield, Nottingham, Oxford) but this post is already quite long enough!
I hope it helps you make a decision. :-)

hi Elizabeth - not making it easy are you? no hints about what your interests are so going on the multiple cathedral cities you're visiting I have to guess that's it so how about routing through coventry (bombed to bits in WW2 and a brand new cathdral as well as being a city that is re-inventing itself (day trip from there to stratford, leamington spa or warwich perhaps?).
or go east to west and visit lancaster - this takes you all the way across the north of england (more than an hour by bus)
nottingham is also en route back south (forget the forest!) its a university town with lots going on and some good museums and splendid architecture.
good luck