One of my favorite places to visit is London. There is so much to see and do there. It has an amazing history, its really easy to get around, there's lots of beautiful parks and it makes for a good base city. There are many other places you can travel to easily and inexpensively from London.
There are also a lot of hidden gems around the city. Whenever I go to London I always walk around the neighborhood I am staying in and walk around the city and are often pleasantly surprised by what I find. I have come across old churches that have cemeteries with graves that date back hundreds of years. Small museums such as the Benjamin Franklin House where your escorted through the house by a period costumed guide in character who tells you about Franklin's life when he lived in the house. Not to mention some great little shops.
So I always recommend to people to walk around the city and now there is a really nice guide to help plan your walks, 33 Walks in London That You Shouldn't Miss by Nicola Perry. This 368 page guide suggests self-guided walking tours around London. Each walk states the distance and best time to walk. It makes landmark and restaurant suggestions as well. The book includes maps and plenty of pictures. You can get the guide on Amazon for under $15.00
Walking the city is the best way to get to know it. It is easy to navigate and the people are friendly and willing to help if you need it. The underground, also known as the Tube, (subway) is great form of public transportation and is very easy and safe to use. So you can easily move from one area to another and begin a new walk.