1: When logged-in to your Gmail account via your favorite browser, in the search box up top, search for:
2: Click the down-arrow in the "select" button, then choose "all."
3: The emails on the page you're on should now be checked; however, to select all of your unread emails, you need to click where it says "Select all conversations that match this search." It will be centered, just above the top-most email, as pictured below:
4: Finally, click the "More" button, then click "Mark as read."