Begin by creating a Google Cloud Project. Navigate to the Google Cloud Console, sign in, and create a new project. Note down your Project ID as it will be required for further steps.

Once your project is set up, enable Firestore by selecting Firestore in the left-hand menu and clicking on "Create Database." Opt for "Start in Native mode" because it provides more flexibility for data operations.

Download and install the Google Cloud SDK on your local machine. This will allow you to use `gcloud` and `firebase` command-line tools for deploying applications and interacting with Firestore.

Install Python and set up a virtual environment for your project. Use `pip` to install the necessary libraries:
```bash
pip install requests google-cloud-firestore
```
`requests` will be used to fetch data from CoinGecko while `google-cloud-firestore` is necessary for interacting with Firestore.

Use the CoinGecko API to fetch cryptocurrency data. CoinGecko provides a free API for accessing data on various coins. Here is a basic example to get coin data:
```python
import requests

def fetch_coingecko_data():
url = 'https://api.coingecko.com/api/v3/coins/markets'
params = {'vs_currency': 'usd'}
response = requests.get(url, params=params)
if response.status_code == 200:
return response.json()
else:
raise Exception(f"Error fetching data: {response.status_code}")
```

Create a Firestore client in your Python script. You will need to authenticate using your Google Cloud credentials. Set up application default credentials by running:
```bash
gcloud auth application-default login
```
Then, use the following code to initialize the Firestore client:
```python
from google.cloud import firestore

def initialize_firestore():
# Use the project ID obtained from your Google Cloud Project
return firestore.Client(project='your-project-id')
```

Finally, write a function to store the fetched CoinGecko data into your Firestore database:
```python
def store_data_in_firestore(data):
db = initialize_firestore()
for coin in data:
# Use the coin's id as the document ID
doc_ref = db.collection('coingecko_coins').document(coin['id'])
doc_ref.set(coin)

# Fetch and store data
coin_data = fetch_coingecko_data()
store_data_in_firestore(coin_data)
```

By following these steps, you can successfully fetch cryptocurrency data from CoinGecko and store it in Google Firestore without relying on third-party connectors or integrations.