Should I use Drush or Drupal Console with a new site? Between Drush and Drupal Console, which seems to have similar functionalities, which one should I install?
You should use Drush.
Drush is about to be added to the drupal/recommended-project, see #3159647, while Drupal Console seems to be abandoned and currently is incompatible with Drupal 9 due to mismatched dependencies, see #4251.
Maybe Drupal Console finally accomplished its mission by triggering the modernisation of Drush. And there's no need for two CLIs. Now forces can be joined again.