How do I update configuration/database after a core update?
Updating Drupal 8 roughly goes like this. First in a local instance of your site:


Run composer update drupal/core --with-dependencies to update core. (Updating all dependencies including contrib would be composer update --with-dependencies.)
Now apply database updates by either drush -y updb or by visiting /update.php in your browser.
Next export the config that may have changed now by either drush -y cex or from your Drupal back-end in the browser.
Git commit and push the updated composer.lock file and the updated config files.


Next on your live site you roughly do the same, but now import the config at the end.


Git pull the updated composer.lock and config.
Run composer install --no-dev to update the files.
Run drush -y updb or visit /update.php to apply the database updates.
Run drush -y cim or visit your admin settings to import the updated config that now should match the one you exported locally earlier.


For a more advanced deployment flow, for example flushing cache and activating the maintenance mode, see my other answer on Order of drush commands for automated deployment?