South Kensington
Museums- Exhibition Road
Victoria & Albert-
Obviously, aside from the exhibits in the museum, please, make a point of visiting the café (especially, "The Green Dining Room", designed by William Morris), and the garden.
5 Thurloe Square- the so-called "Thin House" (Across the road from V&A museum)
Natural History Museum
Science Museum
Royal Albert Hall (across the road from Kensington Gardens- Albert Memorial)
Imperial College London (behind Royal Albert Hall)- Brian May (Queen) did his Physics degree and PhD there. Beit Hall was the site of Queen's first ever concert.
More on Queen landmarks in London (many in the area)- link to map
Holland Park- - please do visit Kyoto Garden
David & Victoria Beckham's house- Holland Park (Elton John and Robbie Williams, among others, also live there)
Robbie William's house- 31 Melbury Road
Instagrammable spots/eating out: Churchill Arms
Kynance Mews- very popular among instagrammers, has been a film location in the past (map)