Traditionally I always run screenshots and colors through colorblind check simulators, though I'm sure there's a lot more that I can do.
Now try to use your own website with a text-to-speech (TTS) engine and your keyboard.
No joke, recreating the users situations is often a cheap and easy way to find problems.
For example I like to check a design by setting the network speed limit with the browsers dev tool to 32kbps, to see how long it would take to load on a bad connection.
I'd urge everybody to take a simple unstyled HTML page as a markup. If your page looks bad without styles, you are doing something wrong. If the navigation is cluttered and convoluted in unstyled mode you are doing something wrong.
If your page is less usable than a basic well structured HTML page, go back and make it a basic well structured HTML page and add styles from there.
In the article most of the annoyances boil down to crappy html markup and unclear navigation. These are things that are easy to fix without any special accessibility knowledge and go a long way in improving usability for everyone.