Leicester is the smallest, but it has a very distinctive culture in England because it very diverse. The Diwali celebration there is a big draw.
Liverpool is the biggest city, so it has a lot of music, shopping and museums. It also has direct flights to the Netherlands, which Leicester doesn’t (you’d have to take a bus or train to Birmingham first). Liverpool is a good base for exploring North Wales as well.
Newcastle is middle sized between Leicester and Liverpool. It also has a lot of culture and direct flights to Amsterdam. It’s a little further from anywhere else, which gives it some interesting features.
I live close to Newcastle as well, so that is my bias, but overall I would probably recommend Liverpool as a first choice.