To send a welcome email to new users after they sign up for an Auth0 application using Twilio SendGrid, developers can configure SendGrid and create an Auth0 Action. First, they need to verify their Sender Identity with SendGrid and generate an API key, which they store as a secret with Auth0. Then, they create an Auth0 Action that uses the SendGrid Mail Send endpoint to send the welcome email. The Action is tested before deployment and added to the Post User Registration flow, allowing it to run automatically when a new user signs up. With this setup, developers can send personalized emails to their users without requiring engineering resources, making it easier to manage authentication and authorization in their applications.