Have A Neutral Test Environment
To successfully diagnose each issue, you will generally want to start with a fresh user profile for the browser you are testing. This helps minimize the potential for unrelated issues to complicate the diagnosis or make the reported issue irreproducible (especially from browser extensions and other customizations). For example, on Firefox you can create a profile which resets each time the browser restarts.
Develop Competencies, Have Impact
Pick up an issue from the list and begin analyzing why it happens. The more issues you analyze, the more familiar you will become with browser developer tools, web site coding practices, web specifications, and just how subtle and whimsical web compatibility issues can be.
You could very well end up filing issues on browsers' bug trackers, interacting with their developers and web standards experts. And ultimately, the web will be a better place thanks to your effort.
Provide as much detail as possible during your analysis. Your thought process while diagnosing an issue is as important as your final conclusion. The more quickly the issue can be fully understood, the more quickly its fix may be negotiated by the folks doing site outreach.