Do I need a phone line for broadband?

"If I get a wired broadband solution, do I need a telephone line? And can I use it anywhere in the house or am I stuck to one spot?"

 You need a phone line for proper broadband, right enough.  Your computer connects to the broadband line through a router, which can allow you to plug in directly or can (if it is a wireless router) allow you to connect over a wireless network.  In the case of a wireless router, you can use it anywhere in the house.

Have a look at these other broadband-related articles to give a more complete picture.