I love the architecture of the Barbican - it is so important to its historical context, at a time when London was being shaped for modernity after the war - they also aspired to make it a bold art centre. We need bold, always.
Slightly biased as this is where I live, but Stoke Newington is great place to spend lazy saturday / sunday afternoons getting sloshed with friends. There are 10 + pubs within walking distance who all are great places for one reason or another. I'll pick out the White Hart for its huge beer garden...
Lisson Gallery is one of the most influential and longest-running international contemporary art galleries in the world. Since being founded in 1967 by Nicholas Logsdail, it has championed the careers of artists who have transformed the way art was made and presented.
Need some clean air or just a time-out? Kew Gardens is a short train ride away from the city centre but is provides you with space, greenery and and a small holiday for the lungs. The garden contain tropical glasshouses and interesting architecture, so it is a treat for both body and mind.