I'm amused that people keep recommending places they haven't been to!
The food at The Hartley is decent, but it's a heck of a way from Southwark tube, and the Lord Nelson does very good ice cream but is otherwise nothing special.
IMHO The Anchor and Hope is the best near Southwark tube, but it is expensive and, as others have said, wildly popular, making it hard to get a table. The King's Arms on Roupell St is, as the previous poster mentions, a good place.
Near Sthk tube, I'd go to the Kings,The White Hart (just round the corner from the Kings), or try the Tas bar (with good Turkish food as well) under the arches between Sthk tube and W'loo. In fact, all the above are on a short circuit (5 mins walk). I used to like the Thai food at The Ring , but the jukebox became so obtrusive and awful that I stopped going.