Google Search Console
Once your site is published, you should always register your site with Google Search Console. It gives you a lot of feedback about your URLs, your sitemaps, and the setup of your site. Another big advantage is that you will receive notifications from Google if they find something is wrong, like a hack, old software versions, indexation issues, etcetera. If you don't register, you miss out on all this. And registering the site may help indexing your site much faster.
Register your site
First, create an account. Then you can start registering your first site by submitting your URL. The first step in this process is always to prove that you are the owner of the site. This can be done in a number of ways, either by adding a text-record to your DNS, placing a small HTML-file in the root of your site, or placing a snippet in the head of your pages. It can even be done using your Google Analytics account. Verification by DNS is the preferred way:
You can also use the prefix method, but that is more cumbersome. In that case, you have to repeat the verification for all versions of your site: with / without www and both the http and https versions. Also, subdomains should be registered separately. All this means you end up with multiple versions of your site in your account, but that is ok, see the next step.
Submit sitemap to Search Console
The next step must be to go to the Index > Sitemaps section and submit your sitemap file, which is basically a matter of pointing Google to the location of your sitemap file. The default location is the root of your domain, but if you have a sitemap extension like OS-Map, 4SEO, or Jsitemap, you should look up the location of the file in the extension parameters. Submitting your sitemap can help you to have your URLs indexed more quickly. For large sites, this is much more important than for small sites.
After submitting your site, Search Console will tell you whether the submission was successful and how many URLs are contained:
When you click the three dots, you can check how many of the URLs have indeed been indexed (takes a few days).
Performance in Search
Under Performance, you will find a great source of information about how your site is performing in the Google search results. I use this report a lot. It tells you about the current performance, but you can also find opportunities for further improvement here. Here is how you can use the data:
Note that the list is initially sorted by Clicks. However, if you sort by Impressions, you can sometimes find phrases or pages that offer great opportunities. In the example above (of a Dutch site), you may notice the third record for "likdoorn". Even though my site shows up on 58.669 pages with search results, only 1% of searchers actually click on my site (CTR = 0,01). With an average rank of 7,2, it just ranks low enough to only enough to not receive many clicks. Maybe I can optimize this page to rank a bit higher and receive much more clicks. This is a slightly simplified example, but that is generally how you can find opportunities.
Note that the report has many tabs, so there are many ways in which you can use the report. I advise you to explore the options.
There are many reports in Search Console, but the Page Experience report is one of the most important ones. Having issues here could definitely harm your rankings. It is a report of how your site is perceived by users, and it deals with performance, navigation and usability and more:
There are separate reports for both mobile and desktop. In this case, despite being fine on Mobile, I have some issues on Desktop. You can drill down to see what issues Google finds on your site. Often, they are related to Core Web Vitals, which is a set of properties Google checks your site for. The following image pretty well describes what these terms mean:
There is a lot more. You can check your structured data, breadcrumbs, review snippets, links to your site, shopping data, etcetera. You can also use the address change feature if you ever want to move your site to a different domain, keeping your rankings!