York is best for time, because I'm there a week. Newcastle has a direct line to Carlisle, but I've only got 3 full days there.You can reasonably do a round-trip from York or Newcastle in a day, with time off for sight-seeing.
Carlisle isn't a bad place to stay, and there are few cheap hotels. But the city centre is very lively at weekends!
Is it easy to get to, bearing in mind I'm reliant on public transport?Bolton Abbey is close enough to Skipton, has many walking routes around the moores.