Yes definitely definitely visit Highgate Cemetry, it's fascinating (if that's the right word!) You do sometimes have to book though I think. Brick Lane is also wonderful, the whole of East London is fab for food especially. Kingsland Road for hundreds of authentic Vietnamese restaurants. Victoria Park and Columbia Road flower market on a Sunday, I sometimes cycle from Columbia Road and sit in the park for a few hours. You can also cycle up the canal near there and it's just glorious. Also Prince Charles Cinema for a really cheap film in Leicester Square, you'd never believe it was there and people just walk right by it.
Other than that I'd say ditch the tube map and walk everywhere, you can stumble across fab new places completely by accident.