Begin by accessing the Gutendex API, which provides access to the Project Gutenberg book database. You can do this by sending an HTTP GET request to the API endpoint, which is typically `https://gutendex.com/books`.

Use Python's `requests` library to fetch data from the Gutendex API. Install the library if you haven't already (`pip install requests`), and write a Python script to send a GET request:
```python
import requests

response = requests.get('https://gutendex.com/books')
data = response.json()
```
This code block fetches the data and converts it into a JSON object.

Once you have the JSON response, parse the data to extract relevant information such as book titles, authors, and publication dates. This involves inspecting the JSON structure and accessing the required fields:
```python
books = data['results']
for book in books:
title = book['title']
author = book['authors'][0]['name'] if book['authors'] else 'Unknown'
# Add other fields as needed
```

Organize the extracted data into a format suitable for CSV writing. Typically, this involves creating a list of dictionaries where each dictionary represents a row in the CSV:
```python
csv_data = []
for book in books:
csv_data.append({
'Title': book['title'],
'Author': book['authors'][0]['name'] if book['authors'] else 'Unknown'
# Add other fields as needed
})
```

Use Python's `csv` module to write the prepared data to a CSV file. Open a new CSV file in write mode and use `csv.DictWriter` to write the data:
```python
import csv

with open('gutendex_books.csv', mode='w', newline='', encoding='utf-8') as file:
writer = csv.DictWriter(file, fieldnames=['Title', 'Author'])
writer.writeheader()
writer.writerows(csv_data)
```
This code writes the data to a CSV file named `gutendex_books.csv`.

After writing, verify that the CSV file has been created correctly. Open the file using a text editor or a spreadsheet program to ensure the data is formatted as expected and all fields are accurately captured.

If you need to handle more data than a single API call returns (e.g., for large datasets), implement pagination. Use the `next` field in the JSON response to fetch subsequent pages:
```python
while data['next']:
response = requests.get(data['next'])
data = response.json()
# Repeat parsing and CSV writing steps
```
This ensures all available data is fetched and written to the CSV.

This guide provides a straightforward approach to transferring data from Gutendex to a local CSV file using Python.