Alright, let's get started by setting up a Firebase project. Head over to the [Firebase Console](https://console.firebase.google.com/) and click on "Add project". Just follow the prompts to name and create your project. Easy peasy! Next, we need to enable the sign-in methods. On the left-side menu, click "Authentication" and then the "Sign-in method" tab. Here, you can enable the sign-in methods you need, like Google, Facebook, or good ol' email and password. Don't forget to save your changes! Now, let's add Firebase to your app. Click on the "Project Settings" gear icon in the top left and then "Add app". Choose your platform (Android, iOS, etc.) and follow the setup instructions. Make sure to add the Firebase SDK to your app as per the platform-specific instructions. Time to get our hands dirty with some code! To get an instance of `FirebaseAuth`, call `FirebaseAuth.getInstance()`. This will let you perform various authentication operations like creating a new user, signing in, and logging out.

For example, to sign in a user with email and password, you can use:

FirebaseAuth firebaseAuth = FirebaseAuth.getInstance();

firebaseAuth.signInWithEmailAndPassword(email, password)
        .addOnCompleteListener(task -> {
            if (task.isSuccessful()) {
                // User is signed in
            } else {
                // Sign in failed
            }
        });

Now comes the fun part—designing your custom login screen! Create a screen where users can enter their details to sign up or sign in. Make it look nice and fit the overall style of your app. Once your custom UI is ready, link it with Firebase operations. For instance, when a user taps the "Sign In" button, grab the data from the input fields (like email and password) and call the `signInWithEmailAndPassword()` method, just like we did earlier.

You can also implement a sign-out feature by calling the signOut() method on the Firebase Auth instance.

Don't forget to manage the user's authentication state using AuthStateListener. Here's how you can do it:

FirebaseAuth auth = FirebaseAuth.getInstance();

FirebaseAuth.AuthStateListener authStateListener = firebaseAuth -> {
    FirebaseUser user = firebaseAuth.getCurrentUser();
    if (user != null) {
        // User is signed in
    } else {
        // User is signed out
    }
};

auth.addAuthStateListener(authStateListener);

This way, you can check the user's state whenever the app starts or resumes, and update the UI accordingly.

These steps should give you a solid foundation to implement Firebase Authentication with a custom UI. Depending on your project's needs, you might need to add more steps or customizations. Happy coding!