This is a pretty open-ended question, but here's where I'd start:
- Technology for handling feeds -- WCF Syndication. Also, read and understand the RSS and Atom specs.
- Infrastructure -- depends on your situation. Is it just for you, or a few friends or are you talking about building the next Google Reader? If it's smaller-scale, then look at a hosting solutions like GoDaddy, DiscountASP.NET, etc. (There's hundreds of them.) If you're talking a larger-scale type of solution, look at hosting it in the cloud - Rackspace, Amazon, Windows Azure.
- Where do you get feeds? Pretty much anywhere. Personally, if this site is for other users, let the users enter them in (why be in the business of trying to guess what feeds people would want to subscribe to?).
I think you need to provide more requirements in order to get more solid feedback. Start with looking at WCF Syndication and get a feel for that library in terms of how to programmatically handle RSS and ATOM feeds (both subscribing and publishing). Once you understand that, I think you'll have a better handle on your next steps.
Hope this helps.