Are you looking for where have lunch, (preferably) dinner or just have a drink?
You can try Lancashire Court, some winding and charming streets in the heart of the fine Mayfair. Classicism and smell of the most traditional British people break, in part, on this place.
You can find elegant but informal restaurants. The four ones have a sophisticated style, what remember you where you are. But the way they are hidden in this area makes of they a very pretty spot where book a table for sharing good times (or enjoy a classic film, like in Hush). On the other hand, at Mews you have a “four in one” building: lounge, restaurant, cocktail bar and dinning room. Other options are: Rocket and La Cave. (A tip: make the reservation over the street tables).
As for shopping, New Bond St. is in the edge of this court, Oxford St. few meters away and the luxuries shops of Mayfair in two minutes. But some art and fashion houses are located in this place: Handel House, Matthew Alexander, Gallery 28, Jo Malone, Elemis Day Spa and from time to time a pop-up store appears.
Nice weekend and smile!