Take the underground or bus to St. Paul's Cathedral and walk across the Thames on the Millenium Bridge, a pedestrian-only bridge. The Tate Modern is housed in a converted power station. The lobby area is open all the way to the 7th floor ceiling. Admission is free. Open until 10 p.m. on Fridays and Saturdays.
To me, the most unusual thing about the Tate Modern is the curator's choice in arrangement of the paintings. Instead of arranging them chronologically (or by artist) as most museums do, the paintings here are arranged by theme: History/Memory/Society, Nude/Action/Body, Landscape/Matter/Environment, Still Life/Object/Real Life (containing a favorite piece of mine, the Claes Oldenburg Giant 3-Way Plug -- Oldenburg is known for his giant replicas of everyday objects). While I'm used to seeing all the museum's pieces by a certain artist grouped together with others of his/her genre, this arrangement allows a comparison of how artists through the years approached a similar subject matter. Very thought-provoking.
Another feature I liked was the lounge area with headphones and recorded mini-lectures. You can pick one of about 10 lectures at any of several headphones and hear explanations of the architecture of the building, explanations of the art styles and movements, or critiques of various pieces of art. Also unusual is the fact that there are very few benches in the galleries. Instead, there are stacks of little canvas folding stools that you can take with you and set up whenever you want to stop and study a particular work. All in all, a truly modern modern-art museum.