Have you setup Google Search Console for your website? Below is the link to get started with the setup. This free tool which will give information on how your website is ranked on Google and for which keywords.
You will need to setup a google email address and add a property (your website domain). Your website will need to be verified which can be done using different methods.
It is a great tool to get information on mobile visibility, sitemaps, external links, internal links, performance, queries, pages, devices used, and any errors.
Make sure to also connect your WordPress website to Google Analytics. There are a few free WordPress plugins that can do this for you such as Google Analytics Dashboard Plugin for WordPress by MonsterInsights.
The various search engines update their databases at their own intervals. It can take several weeks for a new website to be listed. I often suggest getting a free Google Webmaster Tools account so you can verify exactly how they see your website, as well as get and give suggestions for better listings.