I have a Premier account with Barclays, which has been excellent for me. I travel extensively and their internet site is easy to use and efficient. I used to Bank with Lloyds, but they lost the plot some years ago, I had a Gold Service account, but couldn't even call the branch directly (call centres in Wales/Scotland). Now I have a personal account manager that I can call or email directly.
I have an account with Nationwide because their web site is also easy to use, and I can use the ATM card abroad with no fee's and inter-bank exchange rates.