Not the most exciting place inside, but a lovely rooftop area which not all that many people seem to know about, 2 mins walk from Old St station. If it’s packed then The Red Lion is just up the road.
Founded in 1828 the museum houses around 67,000 specimens many of which are now extinct or endangered. My favorite is the skull of a Bottlenose Whale from 1860.
The best thing about any charity shop is that you never know what you are going to find. I have a well tested route around London that includes all of my favourite charity shops. This Oxfam has a great selection of books and often turns up some interesting clothes too.
If you want a peaceful day out of the busy city, Highgate doesn't really feel like London. The cemetery is interesting and peaceful, there's a tour of the old half which is definitely worth the money. Karl Marx, among others are buried there.