How do you start up your own website?
You really weren't very specific, we have TONS of really good advice, depending on what you want to do, and what your goals are, take a look at these:
You can use a free web hosting site and get a domain ( url / address ) that is like so..
Or you can pay for hosting from someone on a site like Hostgator.com
If your just starting go to freehostia.com and use their wizard.