I suggest you look at this website. The London Walks Company.
They do guided walks around London for not much money. Look at their London walks. You can choose a walk that covers a certain section of London. Your guide will take you up and down interesting streets, pointing out landmarks and you will see streets you never thought still existed in London, with a medieval look or a Victorian look. They also have "themed" walks, such as a couple of Beatles walks, a Dickens walk, and more.
My favorite covers the area known as "The City" which is includes the Tower of London area and northward (includes the Leadenhall Market mentioned by Claudia). The City also includes the area traveling west from Tower of London to St. Paul's Cathedral, and northward of that, up to The Museum of London. And the area between these locations. The area pictured here, the old City:
They also do day tours out of London. You meet your guide at a train station, hop on board the train, say to Cambridge, your guide walks you around Cambridge at a leisurely pace, then returns to London with you. A great way to take a day trip.
Scroll through this page. Be sure to click on the red bar at the bottom of the page to see more days trips.
Note that to the right of each day trip on that page, they give you the location at which to meet the guide. You pay at that time.
As Claudia said, you have barely scratched the surface of things to see in London.
A good many things can be found by simply taking a walk along the south bank of the Thames River. Walk along the river and you will find Shakespeare's Globe Theatre, a recreation of his theatre. You can take a tour inside and see the stage and seating for the audience as it was in his day. Or see a play there.
Walking along the south bank, you will also see the Tate Modern art museum, a huge building with an industrial look. Lots of great paintings and sculpture to see here.