First you should enable your docker service to start on every boot by using sudo systemctl enable /usr/lib/systemd/system/docker.service.
Then you should start your docker-compose containers with docker-compose up -d.
-d, --detach Detached mode: Run containers in the background,
print new container names. Incompatible with
--abort-on-container-exit.
(from: https://docs.docker.com/compose/reference/up/)
As far as I know containers running in the detached mode should also restart on a reboot.
If you still want to the the logs or if your containers are running and so on you should get familiar with the different compose CLI commands. See here