For me, I take London as the walking city; though the tube (underground train) system is one of the most extensive in the world, there is nothing like walking from Trafalgar square to the parliament side and walking along the Thames river. Everyone knows the famous museums of London; and the best part: they are free. I never thought myself as appreciating the modern art but I fell in love with the modern art museum of London. It is one of the best museums and its open till late on the weekends.
Though my favourite spot in London has to be Trafalgar Square. No matter what time of the day/night you are there, you are never alone. Its amazing how there are always people in this part of the London. This is the unspoken gathering place for everyone. England wins the ruby world cup, any football match and everyone comes to Trafalgar square to celebrate.
If you want to see the celebrities, Leicester Square is the place to be. Most of the weekends, there is at least one big premier show going on. I was around to see the premier of Cold Mountain and Harry Potter (but with the crowd, could only see stars on the big screen). Definitely living in London will not be complete without talking about the pubs. Being a student, university pubs and places around campus were the best and cheapest hangout places.
Covent garden is one of the most fun places to be in the early evening or late afternoon. Basically a food court with small shops, it has a few of the best road performers of London. Some evenings you can hear great jazz music, see some fun magic shows at this place.
One of the most memorable moments in London was visiting the Wimbledon courts. I am a big tennis fan, so seeing green grass courts was like a dream come true (though i could not see any matches, but people say that during the championship, Wimbledon is in its own festive mode).
Quick Tips:
Magazine called "Time out" has most of the local events listing and things happening in London for the entire month.
Best Way To Get Around:
Within central London, the best way to get around is walking (my friends say i can walk anywhere and everywhere). If walking is not your cup of tea, hopping on and off the london buses is very easy. Day bus passes cost only 2.50 quid. The bus system is extensive around central London but tubes are the best ways to go aound zones outside central London.
Getting outside London is also very easy with virgin trains. There are easy connections to almost everywhere in England.