“When a man is tired of London, he is tired of life.” – Samuel Johnson
I had this quote on a poster when I was in high school, and I have never stopped feeling that way about London. Here are some of my favorite things.
Victoria & Albert Museum — If you only have time for one BIG museum in London, this is my hands down favorite. It’s an eclectic mix of exhibits, with something for everyone.
Churchill War Rooms – a must see for WWII history buffs. Be sure to leave plenty of time. There’s a lot to see, and you really get a sense for what it must have been like for Churchill and his staff during the war. While you are in this neck of the woods, you’ll also have great views of the Thames, Big Ben and the Houses of Parliament, and the London Eye.
Tower of London and if you have time, Tower Bridge. If you only have time for one, I would pick the Tower of London, and definitely do the Beefeater guided tour. While you are over in that area, the viewing gallery of The Shard is well worth the price.
Kensington Palace – They have special exhibits plus a great look into the life of Queen Victoria. In nice weather, explore the Gardens. and if you have small children don’t miss the Diana Memorial Playground.
Cathedrals — Westminster Abbey, especially Poets’ Corner and St. Paul’s, definitely worth it to go up into the Dome.
The Sherlock Holmes Museum is a great stop but remember, it is dedicated to Conan Doyle’s Victorian Sherlock not the modern day BBC version. Madame Tussaud’s is close by and the London Zoo is just across Regent’s Park so you could make this a full day.
London’s parks are all lovely. Hyde Park and Speakers’ Corner is well worth your time, , and in nice weather, nothing beats the walk down the Mall through St. James’s Park to Buckingham Palace. 
Totally touristy, a bit hokey but still great fun, especially for kids — the London Dungeon. It’s right next to the Eye and offers combo tickets with Madame Tussaud’s, the Eye and the Aquarium
Last but not least: go to a show. London theatre is the best in the world.
Where to Stay:
Kensington is my favorite neighborhood. Close proximity to many of my favorite places, including Harrods. 
Go-to Eats:
- Fish & Chips or Ploughman’s Lunch — at pretty much any decent pub
- Curry at pretty much any curry house on Brick Lane
- Covent Garden. Just pick a place that looks good, they are all pricey, but the people watching is fun and if you are lucky a busker may be doing a show.
Getting around:
Buy an Oyster Card and take the Tube. Do not bother with the hop-on hop-off bus tours. Do take a London Taxi at some point.