If you want to learn bout the city itself, then you can't do better than the London Museum. It is a very modern building with a youthful, up to date, and appealing design. Inside there are galleries that stretch across the area's entire history, from its ancient past in the Stone Age and Londinuim's founding in Roman times to more modern periods like the sixteenth century, Victorian era, right up to present date.
For a broader glance at history, go to the British Museum, which covers cultures such as the Ancient Greeks, Romans, and Egyptians. After that, you can choose museums specific to your interests, such as transport, film, fashion. The list is practically endless.