You can’t control exactly how your website appears within Google, nor the position that your website holds when a user searches for you or something relevant to you, but you can (in most cases) control the text that is shown when your website comes up.
Using built-in functionality, you can alter the text that is shown on a page-by-page basis. This goes for news, events, etc. as well.
Here’s an example of a Google search result, showing a Page Title in blue, and a Description in black below the Page Title.
Within a page (and a news article, and an event, etc) you will see the below area of functionality when you scroll down to the bottom as you edit the item within WordPress.
By editing the SEO title, and the Meta description you will control the text that Google picks up when building its search results. Sometimes it uses different text and cannot be overriden, as it controls the text based on what is most relevant to the user’s search. If these are filled in but it isn’t picking them up, it sometimes is a waiting game - it can take days or even weeks for Google to recrawl your website to pick up new changes.
