If you want to be a rate tart take account of the fact that banks dont transfer your money instantly. So if it takes 3 working days to clear into you current account, and another 3 to get to the new, better account you have lost at least 8 days interest.
Some banks/BS are better thanothers at telling you about changes in interest rates.
Birmingham Midshires have a habit of bringing out new and better accounts so that they are top of the best buy tables, but you only find out about them if you are looking, they dont tell you. Northern Rock used to do that too, I think they now write to tell you about changes.
I would go for a phone or internet account, they often have the best rates.
Watch out for introductory bonuses, and remind yourself to enquire, when it is ending, what other accounts they have. You should be able to transfer between accounts with the same bank without lost days.