I live in London

For museums go to South Kensington... Natural History Museum, Science Museum and Victoria and Albert Museum are a short walking distance from the tube station... There's always a contemporary exhibition in V&A... There was one on shoes recently, didn't see it though

I personally like NHM.
I'd get on the piccadilly line in South Kensington, get off in Covent Garden.
From there you can take this route (I suggest you do this in the evening, maybe start at like 6 pm):
Covent Garden --> Leicester sq --> piccadilly circus --> regents street --> oxford street
There are loads of cute restaurants in the streets adjacent to oxford street... Also nice places in Covent Garden
Seeing the changing of the guard at Buckingham Palace is something very touristy but very boring for me. Google for times if you're interested lol
If you to go Westminster you can see London Eye, Big Ben, London Dungeons etc. I wasn't a big fan of the Dungeons.
There's also the Shard, the tallest building in Europe - I think you can get on there for a panoramic views for a fee

Haven't done that either, but it's an option!
Hyde park must be fun in the summer but I wouldn't waste my valuable time there.
For transportation, use CityMapper. MUCH better than google maps
There's a shopping centre called Westfield in Shepherd's Bush but again, I wouldn't waste too much time...
Madame Tussdauds is fun if you have the time and money and it's a museum while it has enough 'pop culture' there to keep both you and your dad entertained!