Learn you an assistive technology (AT) for great good
Accessibility ≠ screen readers! But that said, a screen reader is a freely available AT that offers a great perspective on how some people use the technology we design and build.
Google's Web Accessibility course on Udacity is extremely well-done, and among other things, teaches you how to use a screen reader. (Ignore all the advertising about it being for "advanced developers"—you only need basic HTML and CSS knowledge for most of the class, and the parts that require JavaScript knowledge are easy to skip.)
NVDA is an excellent open-source screen reader for Microsoft Windows. The NVDA tutorial by WebAIM pairs well with it.
VoiceOver is an excellent screen-reader built into MacOS and OS X, and it comes with a great built-in tutorial.
Orca is an open-source screen reader for Linux, although I've never used it myself.
Things to try with a screen reader
I am a fancy but bad button
You should visit here for a neat site about kittens or here for information about cats or here for an article about how your cat is making you crazy.
Inclusivity vs. accessibility
Providing a tabular alternative to a line graph that's only visible to screen-reader users may make that graph accessible, but does it make it inclusive?
Sometimes it's very hard to deal with every possible impairment.
For instance, a high contrast ratio can make content more readable for
most folks with visual impairments, but for individuals with
Irlen syndrome
it can do the opposite.
In this case, try to use open standards that allow users to customize your
design via third-party tooling.