The East End is an older area of London, both residential and commercial associated with markets and business. The City refers to the Financial Quarter. Don't be afraid to walk around the streets. You see so much more and distances are often not as far as you might think - especially in the centre of town. Ask people in the street for directions.
People are often happy to help a tourist.
For shorter journeys take a bus if you can. You see more. Avoid travelling in the rush hours. Do read the link pages on Understanding the Tube System. It has lots of useful information.