If I understood right, you're wanting to host your own web site using your own broadband connection. Basically what that tells me is that finding a web host is not the option you're looking for.
If that is the case, I would recommend two things.
1) Registering a domain name and transferring it over to your DNS
2) Registering a free domain name at
http://www.dyndns.org.
I do the same thing that you're doing for an FTP server and a web site that I just use to edit things before uploading to my actual web site in which I pay for. DynDNS will allow you to choose a domain name with various domain extensions, which was very easy to setup with my router since I use a Netgear and in the router info page, there's an option for DynDNS already.
Example...I have the domain name "superjason.kicks-ass.net" registered through DynDNS. Everytime someone from outside of my network types that into their browser, they will be taken to the website I am hosting(don't try that link though, since I run on windows using EasyPHP for that, so it won't show up everytime people try unless I have EasyPHP turned on).
Hope that helps, and if I'm wrong on your intentions, disregard this =)