There are several ways to automatically build container/docker images.
Here are a few examples:

Note: GitHub Actions can also be used with Gitea and Forgejo. ;-)

Docs:

• https://github.com/features/actions
• https://docs.github.com/en/actions
  • create repository variables and secrets:
    • https://docs.github.com/en/actions/how-tos/write-workflows/choose-what-workflows-do/use-variables
    • https://docs.github.com/en/actions/how-tos/write-workflows/choose-what-workflows-do/use-secrets
  • set Workflow Permission: Settings -> Action -> General and then under Workflow permissions: Read and write permissions
  • https://docs.github.com/en/actions/tutorials/create-actions/create-a-composite-action

Actions:

• checkout
• setup-qemu-action
• setup-buildx-action
• login-action
• build-push-action
• dockerhub-description or multiple-registry-description

.
├── .github
    ├── actions
    |   ├── docker-setup
    |   |   └── action.yml
    |   └── ...
    └── workflows
        ├── build_docker_images.yaml
        ├── build_docker_images_withMatrixAndOwnAction.yml
        └── ... 
├── Dockerfile(s)
└── ...

• .github/workflows/build_docker_images.yaml Workflow-File example for build Docker Images (Multiarch)
• Workflow with Matrix and use own (composite) Action for building Multiarch Docker Images:
  • .github/actions/docker-setup/action.yml is my docker-setup action, that can include in Workflows with

    uses: tob1as/docker-build-example/.github/actions/docker-setup@main

    (Note: You can also use the full commit SHA if you don't want to use main.)
    or when you copy the action:

    uses: ./.github/actions/docker-setup

  • .github/workflows/build_docker_images_withMatrixAndOwnAction.yml Workflow-File with Matrix and use own Action.
  • It is used in my GitHub Repository: docker-php

For GitLab use the .gitlab-ci.yml as an example.
More see in Docs.

Note: My examples are outdated, i recommend using GitHub Actions.

Docs:

• Advanced options for Autobuild

Example for advanced options (hooks) and buildx for Autobuild on Docker Hub to build Multiarch Images

.
├── hooks
    ├── pre_build
    ├── build
    └── push
├── Dockerfile
└── ... more Dockerfiles

.
├── alpine.Dockerfile
├── debian.Dockerfile
├── distroless.debian.Dockerfile
└── scratch.Dockerfile

This examples creates a (simple) Go application. It uses multistages and different runtime containers (with and without shell).
For more examples see in my other Repositories!

• Docker Build
• Building Multi-Architecture Docker Images With Buildx
• https://github.com/multiarch/qemu-user-static & https://github.com/tonistiigi/binfmt

More examples can be found in my repositories. Here is a selection:

• tobi312/tools
• tobi312/php
• healthcheck