To register your domain name you will have to visit some domain registrar like GoDaddy or Namecheap and actually register it. Once registered you need to sign up for a free account with us using your domain name (if not done already) and change the name server entries for your domain to our name servers (ns01.000webhost.com and ns02.000webhost.com) which is done at your domain registrar.. DNS propagation will take a while (24-48 hours) and when propagated you will see our default page.
From there on it is uploading your website files to the public_html directory of your 000webhost.com account which will result in your website being displayed instead of our default.php file. To add your site to search engines you have to submit it to them. For exampe Google -> https://www.google.com/webmasters/tools/submit-url