The first step is to get an instance of MastodonApp (your app). Then, the second step is to authenticate the user. Finally, you get a MastodonSession with which you can call all the methods you want.
Two options : if you already registered your app, then just use the constructor. If you did not register your app on the instance, use MastodonApp.registerApp("mastodon.xyz", "MyGreatApp", "https://localhost:8080/", Scope.WRITE) for example.
To login, just call the login method of MastodonApp. It requires a redirectUri (that can be null) among all the redirectUris provided at app creation, and scopes.
This method returns a LoginProcess instance. This class offers two methods : getLoginURL() that gives you the URL you can give to your user, and authorize(String code) that will confirm the login and return a session once the code provided by Mastodon has been returned
You finally get a MastodonSession. It will in the future implement all the methods of the API directly in a proper Java way, but for now it's more than partial. But don't worry, you can still call directly the APIs using the raw() method.