Google using "broadcom wireless linux" gives a lot of promising hits--eg the one on Ubuntu forums. The thing to do is pick any one of the mainstream distros, and install it. (Don't try to set up wireless when you are running from a live CD.)
Once you have Linux installed and working, it will be much easier for us to provide specific help.
If possible, connect the computer with a regular wired connection before installing Linux. Most installers will now automatically detect and configure the connection. This way, you can immediately start using the package manager to install things (including the wireless driver...