Fear not - you are making life a lot more complicated than it needs to be.
Firstly - broadband supplier. There are two types - one uses existing phonelines, ADSL, for which you must have a BT landline, regardless of who you pay the bill to.
The other sort is cable broadband supplied by Virgin. You don't need a BT phone line but do need to be in a Virgin cable area.
Now, if you choose ADSL the modem and router are usually combined.
If you choose Virgin you will need a separate router connected to the modem if you want to connect wirelessly.
When you have chosen your ISP and have broadband installed you can connect in two ways - by ethernet cable, for which your pc is perfectly capable; or wirelessly.
Your pc may be wifi enabled. If it isn't you can buy a wife adapter fron around �6 - like these:
http://www.blueunplugged.com/c.aspx?c=41106
If your pc is going to be situated near to the modem (or modem router) you are far better to use ethernet cable. Faster, and more secure, than wireless.