There are many attraction and tourist places in London. It depends on your preference, what you like to visit. The most attractions are located in the centre, in Zone 1 and Zone 2, so you don't need to spend a lot of time by traveling. These are the recommended ones:
Top attractions
Buckingham Palace — Westminster, zone 1
Big Ben — Westminster, zone 1
Houses of Parliament — Westminster, zone 1
Tower of London — St Katharine's & Wapping, Tower Hamlets, zone 1
Tower Bridge — St Katharine's & Wapping, Tower Hamlets, zone 1
British Library — St. Pancras, Camden, zone 1
London Eye — Waterloo, Lambeth, zone 1
Madame Tussauds — Marylebone, Westminster, zone 1
The Shard (highest view from building) — Southwark, zone 1
Sea Life Aquarium — Waterloo, Lambeth, zone 1
London Zoo — Westminster, zone 2
Air Line Cable Car — Greenwich, zone 2
Squares
Trafalgar Square — Charing Cross, Westminster, zone 1
Leicester Square — Westminster, zone 1
Piccadilly Circus — Westminster, zone 1
Shopping
Oxford street shops — Westminster, zone 1
Regent street shops — Westminster, zone 1
Bond street shops — Westminster, zone 1
Sloane street shops — Westminster, zone 1
Harrods — Knightsbridge, Westminster, zone 1
Harvey Nichols — Belgravia, Westminster, zone 1
Oxford street shops
John Lewis, River Island, Debenhams, House of Fraser, Zara, Next, Marks & Spencer, Primark, ...
Regent street shops
Ted Baker, Burberry, Hugo Boss, Calvin Klein, Lacoste, Tommy Hilfiger, ...
Bond street shops
Rolex, Chanel, Louis Vuitton, Givenchy, Tiffany & Co, Ralph Lauren, ...
Sloane street shops
Prada, Armani, Versace, Dior, Gucci, Dolce & Gabbana, ...
Markets
Covent Garden Market — Covent Garden, Westminster, zone 1
Camden Lock Market — Camden Town, Camden, zone 2
Greenwich Market — Greenwich, zone 2
Museums
British Museum — Bloomsbury, Camden, zone 1
Natural History Museum — South Kensington, Kensington and Chelsea, zone 1
Science Museum — South Kensington, Kensington and Chelsea, zone 1
Victoria and Albert Museum — South Kensington, Kensington and Chelsea, zone 1
Museum of London — Barbican, City of London, zone 1
Imperial War Museum — Southwark, zone 1
Victoria and Albert Museum of Childhood — Bethnal Green, Tower Hamlets, zone 2
Royal Air Force Museum — Colindale, Barnet, zone 4
Museum of London Docklands — West India Quay, Tower Hamlets, zone 2
National Army Museum — Kensington and Chelsea, zone 1
Bank of England Museum — City of London, zone 1
National Maritime Museum — Greenwich, zone 2
The Royal Academy of Music Museum — Marylebone, Westminster, zone 1
The Sherlock Holmes Museum — Marylebone, Westminster, zone 1
London Transport Museum — Covent Garden, Westminster, zone 1
Galleries
National Gallery — Charing Cross, Westminster, zone 1
Tate Modern — Bankside, Southwark, zone 1
Guildhall Art Gallery and Roman Amphitheatre — Guildhall Yard, City of London, zone 1
National Portrait Gallery — Charing Cross, Westminster, zone 1
Tate Britain — Millbank, Westminster, zone 1
Serpentine Galleries — Kensington Gardens, Westminster, zone 1
Opera & Theatre
Royal Opera House — Covent Garden, Westminster, zone 1
Shakespeare's Globe Theatre — Southwark, zone 1
Sports
Arsenal — Holloway, Islington, zone 2
Chelsea — Fulham, Hammersmith & Fulham, zone 2
Tottenham — Tottenham, Haringey, zone 3
Crystal Palace — Selhurst, Croydon, zone 4
Watford — Watford, zone 8
Wimbledon — Wimbledon, Merton, zone 3