Though it is far from the only assistive technology that people use, screen readers are the most talked about ones in the digital accessibility world.
People who are blind or have low vision use screen readers, and sighted people who find personal benefits from screen readers use them too.
Testing your website content with screen readers can help identify accessibility errors in the websites you build. If a screen reader announces something in a weird way, or if it doesn't announce anything at all even though it should, you can take note of what happened and investigate the steps needed to fix it.
Start with screen readers that are available to you
The best way to start testing your websites with screen readers is to start with the ones that are available to you.
If you're using a Windows computer, Narrator is the screen reader that's built into it.
The most recent screen reader user survey put out by WebAIM shows that JAWS and NVDA are the most highly used screen readers. Both are Windows-based. NVDA is free and JAWS is not, so your budget will dictate if JAWS is available to you.
Voiceover is the built-in screen reader for both Mac computers and iOS devices.
TalkBack is the screen reader that's built into Android devices.
Whatever devices you have available to you, start with testing those screen readers. There are plenty of tutorials out there that you can watch to learn how to test with them.
One thing to watch out for is that if you're not a native screen reader user, then you don't have the lived experience to know the entirely of how native users use this technology. The main reason that you're testing with a screen reader is to find the big obvious bugs that you can detect and fix.
Once you've done that, then your company, if it has the money and inclination, can pay disabled user testers to test your websites and give you information about bugs that you wouldn't be able to find as a non-disabled person.
Start with what you have, learn, go forth, and test!