Hi there!
If you are using the official docker-compose, as per the docker-compose.yml file, volumes with persistent data are stored on /var/lib/docker/volumes/taiga-docker_* or wherever you configured your docker volumes to be (that said, you could map the volumes in the docker-compose.yml file wherever you want them to be).
In the case of the database, the software itself runs on the postgresql container, and the data lives on /var/lib/docker/volumes/taiga-docker_taiga-db-data/.
Regarding backups, do you mean project backups? Database backups? We do not provide neither of those automatically, so in your old setup you probably had some cron script or something doing the job.
For backing up the database, please check this post.
Best regards!
