Fortnum and Mason is a wonderful place. I often go there to get my tea and to look at all the beautiful teacups and other lovely things they have there. It is also a great place to have afternoon tea! I could spend hours in there just walking around, smelling different types of tea, looking at people and thinking about my dream house full of tea cups. The closest tube stations are Piccadilly Circus or Green Park, and it is located on the south side of Piccadilly.
La Fromagerie is a café and cheese shop just off Marleybone High Street. It has great atmosphere and a lovely selection of cheese, wine and food. It is the perfect place for lunch! You could for instance order a cheese platter to share with a friend, or a sandwich or salad...
Foyle's books on Charing Cross Road, near Tottenham Court Road is my favourite book shop in London (apart from Daunt's), and is a must to visit if you are a book lover. It also has one of my favourite cafés in it; Ray's Jazz. The shop has five floors full of books, in every genre you could possibly imagine. I regularly spend hours and hours in there, and would have no problem spending my entire savings in one day. (Thankfully I haven't.) They also have music scores and a record shop that only sells jazz records!
Fulham Palace is dated back to the 11th century and used to be the residence of the bishop of London. It is surrounded by a big park which is lovely to walk through, and it has a nice café on the ground floor. And the best thing is that it is FREE to visit!