Travellin_man is indeed correct, cheap tickets can be had for around $30. Another website to checkout is www.virgintrains.co.uk.

Birmingham itself is an ok city, but nowhere near as fun imho as say Manchester but then it is closer to London. And if you want to look around, it is close to Shakespeare country as well as full of Industrial heritage like the first Iron bridge in the world....but that's another story.

Personally, if I was travelling to London and had time to spare I would do Paris or Amsterdam.