Need suggestions on where to go in the UK.

Okay, I have in mind three areas I want to travel to, and they are all within the UK. England, Scotland or Ireland. Now, where in any of those three places should I visit (city/locale)? Why? What are a few things you would suggest doing/seeing while there, where would you suggest staying (hotel), and is it best to rent a car or use public transportation in that city/locale? My plan is for a visit of 4-5 days.

Edit: I'm the type that doesn't much care for touristy places, more into trying to see/feel the culture/history, etc., of an area.

Edit 2: Been to London, so not interested in that. Also, I'm mostly of Scottish heritage and am thinking I might like Scotland.