Tate Modern
Begin at the former Bankside Power Station, now one of the world's most visited modern art galleries — and it's completely free. The Turbine Hall hosts colossal installations that change annually. Permanent collections include Warhol, Picasso, Rothko, and Hockney. The 10th floor viewing gallery offers sweeping panoramas of the Thames and St Paul's Cathedral.
The viewing gallery on Level 10 of the Blavatnik Building is free and less crowded than the paid viewing platforms across the river. Go at sunset.