On the badges page - if you notice - it starts off with the “Getting Started” badges… the first badges one would normally receive. Then it moves on to the “Community” badges, then “Posting” and finally the “Trust Level” badges. That’s a fairly accurate way to list the badges.
While the badges are easily accessible through the hamburger menu, if you want to make them more visible to users, why not have a header or footer banner with the link to that page YourForumURL/badges? If that would be too noisy for all topics, you could always use the link to the page as a signature line in your posts/replies.
But, this is getting slightly off-topic. 