The Tower Bridge seems like it should be much older than it really is. The bridge opened up in 1894, just a bit over a century ago, using some high tech engineering engines to raise and lower the bascules in under a minute.
Inside of the tower bridge is an exhibit called the Tower Bridge Experience. After paying an entrance fee (£5.5 for adults), you go up one of the towers and can see pictures, films, and artifacts on the history of the bridge. This also includes the original Victorian engine rooms.
In the top level, there are two separate walkways that visitors can go through. Each walkway faces a different side of the River Thames and the London skyline. Pictures can easily be taken from small window holes located at regular intervals. There are also telescopes that can be used to get close ups of various sites as well.
To get some good pictures of the Tower Bridge, it is best to go a few hundred yards over to either side of the bridge. Some of the best pictures will come up near sunset when the lights on the bridge come on. Aside from the Golden Gate bridge in San Francisco, the Tower Bridge is probably one of the most photographed bridges in the world.
It is best to combine a visit to this bridge with the Tower of London, since they are within a couple of minutes from each other by foot.