Start by obtaining access to The Guardian API. Register for an API key on The Guardian's website by creating an account on their developer portal. Once registered, navigate to the API section to obtain your API key.

Use a scripting language like Python to make HTTP requests to The Guardian API. Utilize libraries such as `requests` to send GET requests to the API endpoints. Ensure you include your API key in the request headers to authenticate. For example:
```python
import requests

api_key = 'your_guardian_api_key'
url = 'https://content.guardianapis.com/search'
params = {
'api-key': api_key,
'section': 'technology',
'page-size': 10
}

response = requests.get(url, params=params)
data = response.json()
```

Once the data is fetched, parse the JSON response to extract relevant fields. You might extract fields like `title`, `url`, `sectionId`, and `webPublicationDate`. Structure this data into a format that can be easily inserted into Weaviate, like a list of dictionaries.

Deploy a Weaviate instance locally using Docker or on a cloud platform if you prefer a managed service. The basic Docker command for local deployment is:
```bash
docker run -d --name weaviate -p 8080:8080 semitechnologies/weaviate:latest
```

Define a schema in Weaviate to accommodate the data from The Guardian API. You will need to create a class (e.g., `GuardianArticle`) with properties matching the extracted fields. You can use the Weaviate console or send a POST request to the `/v1/schema` endpoint.

Programmatically insert the structured data into your Weaviate instance. Use HTTP POST requests to the `/v1/objects` endpoint. Here's an example using Python:
```python
import json

weaviate_url = 'http://localhost:8080/v1/objects'
headers = {'Content-Type': 'application/json'}

for article in data['response']['results']:
weaviate_object = {
"class": "GuardianArticle",
"properties": {
"title": article['webTitle'],
"url": article['webUrl'],
"sectionId": article['sectionId'],
"publicationDate": article['webPublicationDate']
}
}
requests.post(weaviate_url, headers=headers, data=json.dumps(weaviate_object))
```

Verify that the data has been inserted correctly by querying Weaviate. Use the `/v1/graphql` endpoint to perform a GraphQL query that retrieves your recently inserted data. Validate the data integrity and ensure all fields are accurately stored.

This guide provides a straightforward approach to extracting data from The Guardian API and inserting it into Weaviate using direct API requests and basic programming techniques.