A browser is simply a program on your computer, just like any other. I can't see how BT could stop any program from working!
What you've got a the minute is just the standard Internet Explorer browser, that comes with Windows anyway, which has simply been 'relabelled' as a BT program. I'm confident that it will continue to work when you change your ISP.
I suspect that the £1.50 per month charge is what you'll have to pay if you want to keep your current BT-Yahoo email address. (Your new ISP will provide you with a new address with their own service, and/or you can get a free email account elsewhere - just ask if you need to know how to do that).
Firefox has a search box at the top right. See the picture here:
http://www.currentnew.../2011/05/firefox5.png
By default the search uses Google but, by clicking on the symbol just to the left of where you enter your text, you can switch to other search engines (such as Yahoo or Bing).
With Chrome, you simply enter your search term in the address bar (where you'd normally type 'www.something' and click 'Go' (or hit 'Enter'). That searches Google. (Incidentally, you can do that with other browsers, such as Firefox, as well).
As we've said before, you can as many browsers on your computer as you like, so there's no actual 'transfer' involved. (You could use Internet Explorer on Monday, Firefox on Tuesday, Chrome on Wednesday, etc if you wanted to. But, if you were dissatisfied with one browser you could immediately switch to any other - you wouldn't even need to close the first browser!).
Try Firefox first. I suspect that you'll quickly grow to love it and that you won't want to use anything else.
Chris