Badges are there to encourage experimenting with features and reward great content. They are a substitute for the user manual.
Groups are 100% about security, if you need a particular group of users to see content that nobody else can see, you start with a group. Then you assign the group to a category.
There is a trust level explanation topic in the faq category here, but in general, trust level lets users have more community cleanup and moderation powers as they spend more time on the site.