Alright, before we dive into the integration, let's get everything set up properly.

• First, download and install Node.js. If you already have it, make sure it's version 8.0 or higher.
• Next, install the Firebase CLI globally. Just run this command in your terminal:

npm install -g firebase-tools

Now, let's get Firebase up and running on your local machine. Here's what you need to do:

• Log in to Firebase using your Google account. Just type this command:

firebase login

• Navigate to the directory where you want your project. Create a new directory and move into it. Then, initialize a new Firebase project with:

mkdir myFirebaseApp
cd myFirebaseApp
firebase init

• During the setup, make sure to select Firestore and Functions.

Alright, with the Firebase CLI and your local project set up, it's time to configure Firestore and serverless Functions in your code:

• Inside the "functions" directory, you'll find the "index.js" file. Open it up.

• Import the Firestore library by adding this code:

const admin = require('firebase-admin');
admin.initializeApp();
let db = admin.firestore();

• Now, let's create some serverless functions. You'll want these to interact with your Firestore database. Here's an example:

exports.createUserRecord = functions.auth.user().onCreate((user) => {
  return db.collection('users').doc(user.uid).set({
    name: user.displayName,
    createdAt: new Date(),
    email: user.email,
  });
});

This function will create a new document in the 'users' collection every time a new user is created via Firebase authentication.

Once you've set up Firestore with your serverless functions, it's time to deploy your app.

• Use this command to deploy:

firebase deploy

Before you start testing your app with the functions you've created, you'll need to set up Firebase Firestore on the Firebase console.

• Go to the Firebase console, navigate to the Firestore Database section, and create a new database.
• When initializing the Firestore database, set the ruleset to Free, so anyone can read or write to the database.
• To test the function, create a new user from the Firebase Authentication section.
• Check if a new document appears in the 'users' collection in Firestore. If it does, your test case is successful!