Not the most exciting place inside, but a lovely rooftop area which not all that many people seem to know about, 2 mins walk from Old St station. If it's packed then The Red Lion is just up the road.
The AA is Britain's oldest school of architecture, but anyone can take advangte of the great work going on there. You can catch really interesting public lectures and free exhibitions throughout the year. There are also occasional installations and event launches, a bijoux cafe and brilliant bookshop. A great place to hang out with the builders of the future.
The oldest public art gallery in London also is my local gallery in my neighbourhood. Opposite to the Dulwich Park where I walked with my puppy regularly. They have great collections of Dutch paintings as well. My husband found the original Dutch painting of the print at his home he was growing up with.