This can be a difficult topic to discuss since it's hard to objectively define trust and authority.
With that said, we will discuss several elements that will give you an increased chance at credibility and authority.
For some perspective, a great question to ask is: "would I enter my personal information into this site?"
No matter what kind of website you're operating, this simple question will immediately provide you with improvements.
Another great way to build trust and authority is by taking care of the small details. If you're scrolling a site littered with typos and filled with sketchy links, would you put your credit card information in?
Of course not.
So take a look through your site and clean up the more subtle errors.
While these changes may not manifest themselves in search results, they will have an impact on your UX.
And if you take care of your UX, customers will be far more open to buying from your brand.
This means that your company/brand is a universally recognized source of accurate and useful information.
Amazon is a trusted authority.
Your neighbor with a 2-week old blog is not.
Part of this equation is time-based.
If you produce top-notch content for months on end, people will take notice. They will interact with your content, share it, and maybe even link to it on their website.
This snowball builds and builds until you have an authoritative website that Google trusts to give the right answers to users.
The second part of this equation is great backlinks.
By continuously producing excellent content, you'll entice others to link to it. And if those other websites have great authority, your website will have increased trust and authority.
Because great brands only link to other great brands, Google is confident in noting you as a trustworthy source of information.
Note that anyone selling "domain authority" tools is trying to scam you. It's a complete guess on their part.
This is a consequence of creating focused, useful content. When you provide the correct answers to search queries, you will receive more engagement as a byproduct.
There are dozens of metrics used to measure how users are interacting with your content, but some of the most important ones are bounce rate, average session duration, and time on page.
Long story short, create meaningful content wrapped in a great user experience, and your site will receive plenty of engagement.
There is no exact blueprint for how to establish this, but it's obvious that time is a factor.
If you're writing top-tier blog posts with spot-on information for months on end, Google has no choice but to increase your site's reputation.
As with authority, Google will increase your reputation when it sees other experts linking to and writing about your content.