Nothing to it! Just hit the reply button on comments you want to respond to. You can also upvote anything you like by clicking on the up arrow to the left of the comment.
You're not the only person who misses the old newsgroups! The format that Hacker News uses is one that became sort of standard on the web in the early 2000s. It works differently than usenet did, but you get threaded comments in the sense that replies are nested under the posts they're replying to.
