I lived in London for 5 months and loved every minute of it. One place to go is the Highgate Cemetary. It's located on the north side and has a website you can check out.
I used to pick a tube stop and walk the neighborhood to the next tube stop. You can purchase an "all zone" ticket for the undergound that will allow you to go whereever you want, as many times as you want all day.
Post #11 is for you, lawgirl. ;)