Maybe an obvious choice, but the Barbican is always good. My favourite space in there is the curve gallery, last time I went was for Cory Arcangel's 'Beat the Champ'. I love the architecture, although not many people do. I think it was voted 'London's ugliest building' in 2003.
The closest you can get to Little Italy in the east.
The owners are amazing, the atmosphere is relaxed, the lunches are abundant and homemade.
The cakes are amazing, try Torta Clara (pear and chocolate) for the ultimate bliss.
A wonderful collection of printed material organised chronologically throughout the space. Examples of old newspapers and other printed material give context and meaning for the visual styles.
One of the main sources of inspiration for my work are unusual objects, vintage toys and tools. A great place to find them is this genuine car boot sale, which is clear of 'bargain' toilet paper and questionable cosmetics. Another plus are the comparatively sociable opening hours. (it starts at noon… )