As a developer, when should I test my website?

I know this might be a newbie question but I couldn't find an answer anywhere.

After I release my website I test it in a few browsers. Also, when a new Android device comes out I'll take a look there also. I'm curious about what other milestones I should use to guide my website testing schedule (for example, when a new version of Chrome is launched should I make it a priority to test in that new version?)

Just as background: My site has high quality requirements and needs to work in a lot of different browsers.

Thanks in advance!