The Hayward Gallery in Southbank center has held my favourite exhibitions in London and each time they are extremely well curated. Definitely a place to visit !
The V and A houses a public collection of prints and drawings of every kind imagineable from old master drawings to photographs, playing cards, fashion plates and wallpapers. You have to select what you want to see from the collection http://collections.vam.ac.uk/and book in advance. The behind-the-scenes route up to the study room of v and a is a treat alone (pictured)
Beautiful architecture and setting, ideal place for catching the latest art-house flick or to enjoy evening tea and a journal session. Friendly staff, ambient atmosphere and a wonderful crowd.