There are two ways you can get your website to start showing up on Google search results.

First, you can submit your site to Google using their webmaster tools. Using this tool you can let Google know about your website, claim ownership of the site, and also submit a site map (optional).

Second, Google will automatically index your website as long as other websites are linking to it. If lots of sites link to yours, eventually the Google crawler will stumble upon your site.

If you want your site to rank highly, you should pursue both of these options.