The move to Manchester sounds very exciting. I spent 3 years there as a student, and it really is a great city.
I'd recommend Didsbury as a potential area for you to settle down with the family. You can get ideas by reading our Areas and Suburbs in Manchester page.
What kind of school do you have in mind for your children?
You should also bear in mind that there are some excellent comprehensive schools too, which are free. Of course there are also some very poor ones, but you can check the Ofsted reports and the exam results to get a good idea of what they are like. Also try looking at mumsnet.com.
Yes, you are right fees at private schools in the UK are high. If at all possible it is best to negotiate an allowance in to your contract of employment to allow for this expense.
Catholic schools are a great option and there are quite a few to choose from in Manchester such as St Matthew's, St Paul's and St Peter's. However, waiting lists can be long, so I'd apply as soon as possible to get the ball rolling.
Do you think you'll have the option to do a visit to Manchester with the kids so they can get a feel for the schools before making the move?