London Transport Museum
It is a pretty nice museum to visit. You do have to pay £15 to get in, but the ticket is valid for a year so you can come back as much as you want in that year. Exhibitions change pretty regularly too so it's good value. Indeed it's a pretty great place for kids and has a lot of interactive elements, including simulations of the drivers' controls.
The museum takes you through the history of public transport in London, and you can sit in a lot of the replicas which is really interesting. It takes you through the present and even to future developments of the transport system, if I recall there was some interesting information on Crossrail and sustainability. It covers London from around 1700 when horse drawn carriages were all the rage through its expansion and the introduction of the tube to modern developments like Crossrail. You can see how London grew as transport improved over the years with a doubling of size between the two wars. As a daily user of the Northern Line it was good to sit in old tube carriages and see how they have got better over the years (hard to believe I know). The museum catered perfectly to my inner geek and I have no hesitation in recommending it.
Did you know that London had a subway system before they even had electricity? I didn't, until I came here. There are old subway cars you can sit in, and you can see the natural progression of the subway system, with line additions through the ages. They even have a simulation for running the trains! If you are a transit nerd, this is a must-see.