Begin by setting up an Amazon Redshift cluster. Log in to the AWS Management Console, navigate to the Redshift service, and create a new cluster. Choose the appropriate node type and number of nodes based on your data size and query performance needs. Ensure that the cluster is launched in a VPC with appropriate security group settings to allow incoming connections from your client machine.

Configure your Redshift cluster's security group to allow inbound traffic on the port Redshift uses (default is 5439). Set up an IAM role with the necessary permissions to access Redshift and add it to your cluster. Make sure to also configure network settings to allow your client machine to access the Redshift cluster, either through public IP or VPC peering.

Sign up for an account on Polygon.io and obtain your API key. The API key is necessary to authenticate requests to the Polygon Stock API. Keep this key secure as it provides access to your API data.

Create a Python script (or use another programming language of your choice) to extract data from the Polygon Stock API. Use the `requests` library in Python to send HTTP GET requests to the API endpoints. For example, to get stock prices, you might use:
```python
import requests

api_key = 'YOUR_API_KEY'
url = f'https://api.polygon.io/v1/last/stocks/{ticker}?apiKey={api_key}'
response = requests.get(url)
data = response.json()
```
Parse the JSON response to extract the necessary data fields for your Redshift tables.

Transform the extracted data into a format that is compatible with Redshift. This often means converting JSON data into a CSV format. Use libraries like `pandas` in Python to clean and transform the data:
```python
import pandas as pd

df = pd.json_normalize(data)
df.to_csv('data.csv', index=False)
```
Ensure that the CSV file matches the schema of your Redshift table.

Upload the transformed CSV file to an Amazon S3 bucket. Use the AWS CLI or Boto3 library in Python to perform the upload:
```bash
aws s3 cp data.csv s3://your-bucket-name/data.csv
```
Make sure the IAM role associated with your Redshift cluster has permissions to access this S3 bucket.

Use the `COPY` command in Redshift to load data from the S3 bucket into your Redshift table. Connect to your Redshift cluster using a SQL client, and execute:
```sql
COPY your_table_name
FROM 's3://your-bucket-name/data.csv'
IAM_ROLE 'arn:aws:iam::your-account-id:role/your-redshift-role'
DELIMITER ','
IGNOREHEADER 1
CSV;
```
Adjust the options in the `COPY` command to match your data format and Redshift table schema. This step efficiently loads your data into Redshift for further analysis.

By following these steps, you can successfully move data from the Polygon Stock API into Amazon Redshift without relying on third-party connectors or integrations.