To get started with Contentful, users can create a new space in the web app, which can be an empty space or an example space that comes with a content model and sample entries. The user then connects their web project to Contentful by adding a CDN script tag to the HTML file and opening the `contentful.js` file to authenticate and fetch content from Contentful. The user can see how Contentful works by rendering example entries on their web page, exploring the relationship between content types and linked reference fields, and modifying the product entries to test different scenarios. With this foundation, users can start building their own projects using Contentful and learn about its capabilities for delivering fast, organized, and quick build-to-ship processes.