I live in Beverley, and I love it.
We moved from Hull 18 months ago into a rented property as we were having issues with DDs school, she leaves in a year but we like it that much we're buying here.
Parking in the town centre is a pain, but most of the nice or nice-ish areas have off street parking anyway. Wherever you move to inside Beverley will be less than a 20-30 minute walk to the town centre anyway, with buses. The parking in the centre is suspended every Saturday as they have a market, and often Sundays too as they put on very regular events. Don't let that put you off though, there's so much to do here!
Trains to Hull run twice an hour, which then connect to Leeds/Manchester/London etc. The Leeds/ Hull train is almost exactly 1hour, that train sometimes connects to Beverley, but more often than not would be connecting. However, as I say, they're twice an hour and take less than 10 minutes, or buses run very regularly too. You're less than an hour away from Leeds on the motorway driving too, a very easy drive.
Walkington is a lovely quiet village, as is Cherry Burton and many others around the area, but transport links obviously are not as good, you would be reliant on cars. I could manage quite easily without a car in Beverley.