So, what would I have to do to connect myself to the internet(not through and isp but my own computer), so I dont have to worry about downtimes and if I can do that how do I host a site from my connection?

You need an ISP. You are not going to be able to join the Internet, and certainly not host a site, without one. At best you could set up some sort of VPN that would allow other specific machines to connect to yours if they are configured properly but then that's not much use if you want people to actually access your site.

You could contact a company such as this:

http://yipes.com/services/internet-access.html

 

But keep in mind, there would be a large investment, both in monthly service, and in the purchasing of a router, switch, etc.

That's an ISP.