Begin by familiarizing yourself with the SpaceX API documentation. The API provides endpoints for various data points like launches, rockets, payloads, etc. Identify the specific data you need to move to Snowflake and note the endpoints and parameters required for accessing this information.

Prepare your development environment by installing necessary tools. Ensure you have a programming language runtime installed, such as Python, which will be used to make API calls and process data. Additionally, ensure you have access to the SnowSQL command-line client for interacting with Snowflake.

Write a script in Python to fetch data from the SpaceX API. Use the `requests` library to send HTTP GET requests to the API endpoints. Handle the response data, which is typically in JSON format, and parse it to extract the required information.

```python
import requests

response = requests.get('https://api.spacexdata.com/v4/launches')
data = response.json()
```

Transform the JSON data into a format suitable for loading into Snowflake. This may involve flattening nested JSON structures and converting them into CSV or another tabular format. Use Python's `csv` module or Pandas to handle this transformation.

```python
import csv

with open('spacex_data.csv', 'w', newline='') as csvfile:
writer = csv.writer(csvfile)
# Write headers
writer.writerow(['id', 'name', 'date_utc'])
# Write data
for launch in data:
writer.writerow([launch['id'], launch['name'], launch['date_utc']])
```

Log into your Snowflake account and create the necessary database, schema, and table to store the SpaceX data. Use the Snowflake web interface or SnowSQL to execute the SQL commands.

```sql
CREATE DATABASE spacex_data;
CREATE SCHEMA spacex_data_schema;
CREATE TABLE spacex_data_schema.launches (
id STRING,
name STRING,
date_utc STRING
);
```

Use SnowSQL to load the transformed data into Snowflake. First, upload the CSV file to a Snowflake stage, then copy its contents into the target table.

```bash
snowsql -q "PUT file://spacex_data.csv @%spacex_data_schema.launches"
snowsql -q "COPY INTO spacex_data_schema.launches FROM @%spacex_data_schema.launches FILE_FORMAT = (TYPE = 'CSV', FIELD_OPTIONALLY_ENCLOSED_BY = '\"')"
```

Finally, verify that the data has been correctly loaded into Snowflake. Run a SQL query to inspect the data and check for consistency and accuracy.

```sql
SELECT * FROM spacex_data_schema.launches;
```

Review the output to confirm that the data matches what was fetched from the SpaceX API.

By following these steps, you can effectively move data from the SpaceX API to the Snowflake Data Cloud without relying on third-party connectors or integrations.