"How do I set Gmail as my browser's default email client for mailto links?"
SHORT ANSWER: Sadly, there is no short answer. Let's review how to update the settings.
INSTRUCTIONS FOR CHROME: We highly recommend the Chrome browser to enable all the superpowers of G Suite. If you use another browser, please check here for the full instructions.
Step 1: Open Gmail in Chrome and click the Protocol Handler icon in your browser's address bar. If you don't see this icon, skip ahead to step 3.
Step 2: When prompted to "Allow Gmail to open all email links?", select "Use Gmail," and click "Done." Then you're done! Skip steps 3-5.
Step 3: If you were unable to see the Protocol Handler icon , click the Chrome menu icon in the top right corner of your browser, and choose "Settings."
Step 4: Click the "Show Advanced Settings" link at the bottom of the screen. Then click the "Content Settings" button under the "Privacy" header.
Step 5: In the pop-up window, scroll down to the "Handlers" section and click the "Manage Handlers" button. Then select Gmail from the mailto dropdown, click "Done," and, well, you're done!