I made a new site, what should I do to remove the old site from Google search result and put the new one in?

I have very limited knowledge about SEO. The problem I am facing is that after my new website built with wordpress went live, when I search for my website’s name, my old website that was built with Wix still appears as the first results, while my new website is not even in the first page.

What is the best practice to promote the new website? Please note that my old site has been around for a long time. Instead of simply shut down the old website and start a new SEO campaign for the new website, is there a way to have my new website inherit the reputation the old one already has with Google.