To get started with Go modules and Cloudsmith, one can create a new Go module by defining a `go.mod` file. A private Cloudsmith repository can be created to host the module, which allows for secure and version-controlled distribution of the module. Once the module is pushed to the repository, it can be pulled as a dependency in another Go project using the module's URL. This enables developers to share and reuse their code efficiently, while also ensuring that dependencies are up-to-date and secure. By leveraging Cloudsmith, developers can easily manage and distribute their Go modules, streamlining their development workflow.