To get started with Docker and Cloudsmith, build a Docker image by creating a new directory, initializing a new Docker project using `docker-compose`, and defining the application code. Push the Docker image to a private Cloudsmith repository using the Cloudsmith API or CLI tool. Then, pull the Docker image from the Cloudsmith repository and run it on your local machine using `docker run`. This demo showcases how developers can use Docker and Cloudsmith to streamline their development workflow and deploy applications quickly and securely. The process involves defining the Dockerfile, building the image, pushing it to a private repository, pulling it back down, and finally running the application.