Hi SG - a few places to consider visiting. Try the back streets around Borough Market - that's next to Southwark Cathedral on the south and wets side of London Bridge. The area has become a bit trendy these days, but there's still some original spots, like
Clink Street which was the site of London's oldest prison. Walk a little further around the lanes and back twoards the Thames and you'll cone across the very old
Anchor pub, and just along from there is the reconstructed Globe Theatre. Trace your steps back to London Bridge Road and heading south about 100 yards on the left is The George Tavern (can't find a picture) an old coaching inn in a private yard.
Another pub worth visiting is the
Blackfriar on the north side of Blackfriar's Bridge - it's pure English Arts and Crafts Movement design with plenty of beaten copper murals and original features inside. Move into Fleet street and there's the Cheshire Cheese pub (a haunt of Charles Dickens) and also Dr Samuel Johnson's house (just behind Fleet street in an old cobbled courtyard).
Lastly, on a Sunday, try
Spittalfields Market - close by Liverpool Street Station and off Bishopsgate in London EC3 - it has lots of intersting shops and resaturants - and probably the best Tapas bar/restaurant in the city!