To visit as many places in London as possible during a day, you have to start early in the morning, stay mostly in the centre and travel by tube, because it is the quickest way how to travel.
Before the trip, you have to set your priorities, so you have to know, what you want to see definitely and what you would like to see, if you have enough time. For example, there are many impressing free museums in London, but you can spend hours in the largest ones, so you will not have much time to visit other places.
Things to do in London in 1 day:- Buckingham Palace
- Houses of Parliament
- Big Ben
- London Eye
- Tower Bridge
- View from Sky Garden (requires a free booking few days / weeks ago) or 120 Fenchurch street rooftop (no booking is required)
alternatively
- Canary Wharf (travel by DLR)
- View from Royal Observatory in Greenwich
- Little Venice
- Picadilly Circus
- Leicester Square
- Trafalgar Square
- Covent Garden
- Soho
- China Town
Things to do in London in 3 days:- British Museum (visit can take you almost all day)
- Museum of London
- Hyde Park
- Kensington Gardens
- Holland Park
- Churchill Arms Pub (impressing pub, located between Hyde Park and Holland park)
- Harry Potter Platform 9 3/4 at King's Cross Station (may be interesting, if you travel with kids)
- Walk through Thames Underwater Tunnel (between Isle of Dogs and Greenwich)
Things to do in London in 1 week:- Science Museum
- Victoria and Albert Museum
- Natural History Museum
- Bank of England Museum
- London Mithraeum
- Hampstead Heath