Suggestions for a four day trip in London

Where are you staying? That might help with giving you things to see in your "neighborhood"

Are you museum and church kind of folks? Theater folks? Historic Palace folks? WWII history?

There is enough in London to keep you busy for weeks. My choices might not be yours as we all travel differently!

Here's what I love for a first timer:

Tower of London
Westminster Abbey
Churchill War Rooms
National Gallery
Windsor Castle if you want a day out of the city center although with just 4 days I'd only do this if you really are interested!
Kew Gardens if you are a garden fan
St Paul's Cathedral (and you can walk there from Tower Of London via Leadenhall Market where some of Harry Potter was filmed, lol!!)
A walk with London Walks
British Museum
British Library (Magna Carta)
Shakespeare's Globe Theater

Many are in to the food markets and enjoy the Borough Market. Not my thing so no experience there!

These are just the tip of the iceberg! Some need booking ahead (Westminster Abbey, Tower of London and Churchill War Rooms)

Have a great time!

Here are Rick's suggestions:

London sightseeing suggestions -- click on At a Glance

Based on what others have said I wouldn't recommend the Changing of the Guard. It takes far too much time to stake out a position from which you have a good view.

Rick's London at a Glance, under the Explore Europe section, is a good place to start.

https://www.ricksteves.com/europe/england/london

I agree that the changing of the guard isn't worth the time it takes. I would add in an evening at the theatre, either in the West end or at the Globe.

The metro in London is called the Underground or tube.

London is one of the most often visited and most famous European capitals and you are asking what to see?

Besides walking what other interests do you have?

Royals? History? Theatre? Shopping?

IF you are interested in something related to the Changing of the Guard at Buckingham Palace, I like to walk along Birdcage Walk to the Wellington Barracks where the New Guard forms up and are inspected in the forecourt area. The band plays and then they march out the gate and over across the street to Buckingham Palace.

https://goo.gl/maps/ceUEoGnMC73YhQo17

Here is the timing from the official website of the Household Cavalry.

https://www.householddivision.org.uk/index.php?action=changing-the-guard

You can also watch a detachment separate from the main group and march down Pall Mall to St. James' Palace to change there.

Parks in London in the Spring and probably in September are very relaxing and beautiful. Possibly a short or long Thames boat ride to see the city for a different perspective.

If you like the theatre, I'd go see a play. I'm not huge into theatre but I went to the Tina Turner musical and it was fantastic, about 44 pounds. Also, at Leiscter Square they sell day-of theatre tickets at discounted prices. I didn't do that, but I heard it's a great way to go.

Agree with your list, it's great. I'd also walk along the Thames and cross the Tower Bridge by foot. The National Portrait Gallery had a great Paul McCartney photo exhibit going on right now.

I'd avoid the Natural History Museum, last two times I went it's been insanely crowded. One caveat, for a few extra pounds they have the National Geographic nature photos of the year exhibit, but it was sold out when I went this summer. That is not crowed because it costs 15 pounds or so.