The first week in London has not yet ended for me and I already have seen a huge amount of things. One of the best things about London is that you can see a lot of things for free. Catching a tube or bus is easy and for the rest you can get around by foot easily. The amount of walking I did these past days is insane!
In all honesty: The centre itself is actually rather small and you are better off walking a short distance instead of taking the tube.
While the tourist spots are alway buzzing with people the parks are a nice spot to take a breath and calm down.
St. James Park
Saint Paul´s Cathedral
St. James Park, the Buckingham Palace and the Saint Paul´s Cathedral are in a very short walking distance from each other.