Select a good web host and register the domain name with them.
Then upload the site to your new server space. It takes a day or so for newly registered names to be propagated to the DNS servers, and if you've uploaded your site in the meantime it will not be visible. Once the name location has been registered on the DNS databases your site will appear in all its glory.
It's not really necessary to submit to search engines - they'll find you. Don't expect it to happen overnight though, some of the engines take longer than others to get around to crawling your site.