Which statistics like page visitors or most visited pages are tracked automatically, or do I need extra modules for that?
Everything in Drupal is provided by a module (core, contrib or custom), so one way or another you'll need to enable one to get extra functionality.

Drupal core comes with a Statistics module, you can enable that and see if it has what you want easily. If not, just search around for the different statistics modules there are out there, and install and try them.

If you stick to stable releases, you shouldn't have any trouble removing a module that you find doesn't do what you need, so there's no downside to trying.