From your main menu, access superzoom, boomerang or rewind mode. From there, add on text or animated stickers. Once your message is complete, toggle through your contacts list to choose who to send it to, or take it public via Instagram Stories.
Swiping sideways to your right from your main menu opens up a side menu that allows you to access your profile or explore other Instagram content directly from its official app. Otherwise, swiping left opens up your inbox with copies of previously sent and received direct messages. There's also a search option to easily locate a specific message in your inbox.
That said, Directs from Instagram isn't quite as detached from Instagram as it might seem. You'll still need the official Instagram app to make the most of all of its features. But, in any case it looks like old school, permanent social media posts are on their way out as temporary, private posts take over.