For it to happen to both firefox and IE is interesting.
I don't know much about filezilla but it definately looks OK and seems to be still supported very well.
And I'm assuming if you don't upload files all is fine and you can browse the web, watch you tube till your hearts content?
It almost sounds like on certain actions/protocals being used, something is trying to proxy everything sent from that point. (I'm also assuming your network connection is still active when this happens, but since you can ping websites I think we can assume that is a safe bet)
2 things I'd be interested in, one being a hijackthis log (
http://download.cnet.com/Trend-Micro-HijackThis/3000-8022_4-10227353.html )
Hijackthis finds all the things that load on bootup, load into your browser when you start it, stuff like that.
Second is what antivirus/antimalware products do you have installed?
Cheers,
Terry