After mucking up my installation quite well, I decided it was time to reformat and start again. I am sure what I had managed to do was fixable by someone with more Linux knowledge than me but I wanted to get on, so a reformat was in order.
Since reinstalling everything is a lot rosier. It turned out my problem was, was that I had specified the Primary DNS address in the Primary DNS address field - to me this sounds obvious. But, for reasons best known to the place where I work, I had to actually specify the Gateway address as the Primary DNS address and so the Secondary DNS became the Primary. Now I can browse the web and still send and receive emails - perfect!