Docker Hub
Docker Hub is my choice for a container repository. Unfortunately private repositories are a paid option but I think it's relatively cheap so it's worth getting. Once you have a private repository, you can start configuring an API key which will be used with GitHub to generate builds.
Generating An API Key
Start by going to account settings and selecting "Personalized Access Token"
Select "Generate new token"
Configure Access Token with Full Access
Save Access Token
Creating Repositories
Repositories in general will be public but if you have a private repository, you can generate private repositories as long as you select "Private" when you create the repository. This will require authentication later when downloading containers for use in Kubernetes.
Securing Builds
Docker Hub comes with security on your repositories called Docker Scout. You must enable this for each repository you want scanned.
Once you have Docker Scout enabled, you should go to Docker Scout to configure your policies.
Now keep in mind that it Docker Scout only gives you two free live monitoring options without paying for it. If this is not feasible for you, use the static scanning option.
No Unapproved Base Images
Go to Policies > No unapproved base images.
Select Edit
Change from * to docker.io/*
Scout Scores
You should try to achieve good scores on Scout to ensure security