Are you going to be hiring a car?
What time of year would you be visiting?
Canterbury with visits to Dover, Whitstable for a seaside experience and a great oyster festival, Leeds Castle, the Kent Battle of Britain Museum in Folkestone and castles at Walmer and Sandwich.
York - the historical city centre plus the great Rail Museum and visits to either Harrogate or Leeds. North York moors and the Yorkshire Dales for great countryside. Yorkshire's not known as "God's own county" for nothing (and I am a Lancastrian)!
Rutland - like the Cotswolds, but more compact, nicer and far fewer tourists! Visit Stamford, where Middlemarch was filmed and Burghley House; Oakham, a pleasant market town with the unique Oakham Castle's collection of horse shoes (more interesting than it sounds - the Queen has presented them with horse shoes!), take a boat trip on Rutland Water and see Uppingham, another market town. Leicester is about an hour away to see Richard III. One of my favourite parts of the country.
Rye as a base for seeing Bodiam Castle, Battle or Hastings. Leeds Castle, Hever or Chiddingstone castles are also possibilities from Rye.
I am one of those that think the Cotswolds are over-rated and overly busy compared to many other parts of the UK. Stratford upon Avon too is too tourist tacky for my tastes. Arundel is interesting for a day trip, but there is less of interest for days 2 & 3 compared to some areas.
Salisbury could be combined with a day in the New Forest at say Beaulieu and another in Winchester.
Too many choices!