I would recommend getting comfortable using the built-in screen readers on your own devices (VoiceOver, Narrator, etc). Lots of people use those same tools. You'll probably just want to mute the screen reader and use the developer option to display what's being announced on screen as text.

Get comfortable navigating web pages using the various functionalities available (the ability to scan only headlines, for example).

I also open the accessibility tab in the web inspector in Chrome or Firefox to see the accessibility tree, which is like a simplified version of the DOM tree that gets exposed to assistive technologies.