• Stop the server:

sudo systemctl stop docker

• Create/edit the configuration at /etc/docker/daemon.json, for example:

{
    "data-root": "/new/path/docker-data-root"
}

• Copy your data there:

sudo cp -axT /var/lib/docker /new/path/docker-data-root

(if the target docker-data-root directory already exists, make sure you don’t accidentally copy into a docker subdirectory).

• Start the server:

sudo systemctl start docker

• Check everything works:

sudo docker images

The previous answer:

1. Uses an obsolete/undocumented(?) option -g. Instead you should use the option documented in dockerd --help, which is --data-root.
2. Edits the service file, which might cause problems when installing updates. Instead you should edit /etc/docker/daemon.json.
3. Uses rsync to copy files without preserving ACLs, xattrs and hard links, though I’m not sure there’s any.