Exactly how you prepare your app(s) for the upcoming changes will depend on your app/environments, such as whether or not you're using any official Dropbox SDK(s). If you are using an official Dropbox SDK, please make sure you're using the latest version of it, and refer to the current documentation to make sure you're using it as documented.
You can also find more information on the migration for access tokens and scopes in this blog post.
For the new token system, you can can control the behavior, e.g., to get new short-lived access tokens/refresh tokens, via the parameters for the OAuth flow. You can find information on the available options in the authorization documentation. (If you're using an official SDK, please refer to the SDK documentation.) You can also control the default, for when the app doesn't supply the token_access_type parameter, via the "Access token expiration" setting for the app on the app's page on the App Console.
The scope migration is backwards-compatible (just make sure you enable all of the scopes you do need), but you can also register a new API app to test with if you wish. New apps are scoped from the start.