Begin by ensuring that the data you want to move is in a suitable format. You may need to run SQL queries to clean and transform the data into a structured format suitable for export. Use SQL commands in Redshift to create a staging table if necessary, ensuring the data is organized for efficient export.

Use the UNLOAD command in Redshift to export the data to Amazon S3. This command allows you to export data in parallel to a specified S3 bucket. The data can be exported in CSV or Parquet format, depending on your preference and the complexity of your data:
```sql
UNLOAD ('SELECT * FROM your_table')
TO 's3://your-bucket/your-folder/'
CREDENTIALS 'aws_access_key_id=your_access_key;aws_secret_access_key=your_secret_key'
DELIMITER ','
ADDQUOTES
ALLOWOVERWRITE;
```

Once the data is successfully exported to S3, you need to download it to a local machine or a server that can access Teradata Vantage. This can be done using the AWS CLI:
```bash
aws s3 cp s3://your-bucket/your-folder/ ./local-folder/ --recursive
```

Before importing the data into Teradata Vantage, ensure that the target table is ready. Use SQL commands within Teradata to create a table that matches the schema of the data you exported from Redshift. This step is crucial to avoid any data type mismatches or errors during the import process.

Use the Teradata FastLoad utility to import the downloaded data into Teradata Vantage. FastLoad is designed to quickly load large volumes of data into an empty table. This requires preparing a FastLoad script that specifies the input file and maps it to the Teradata table:
```plaintext
SESSIONS 4;
SET RECORD VARTEXT ",";
DEFINE
column1 (VARCHAR(50)),
column2 (INTEGER)
FILE=./local-folder/your_file.csv;
BEGIN LOADING your_teradata_table;
INSERT INTO your_teradata_table (column1, column2)
VALUES (:column1, :column2);
END LOADING;
```

After loading the data, it is essential to verify that the data in Teradata matches the data originally in Redshift. Run count checks and, if possible, checksum or hash checks on both ends to ensure data integrity. You can use SQL queries in Teradata to perform these checks.

Finally, clean up any temporary files, both locally and in your S3 bucket, to avoid incurring unnecessary storage costs. Additionally, consider optimizing the Teradata tables by collecting statistics or running any required maintenance commands to ensure optimal performance.

This guide outlines a direct and practical approach to moving data between Redshift and Teradata Vantage using basic tools and utilities provided by the platforms themselves.