Finding citeable statistics
Is there an advised, reliable way to find and cite the data? Only from
a certain list of websites?

A quick one would be Google Dataset Search.
A more 'community'-led one would be DataCite whose stated aim is to provide DOIs for research datasets.
If you need data owned by the public sector (governmental data), so-called Open Data Portals are the way to go. For instance, this one by the EU.
However, the whole area of research data management (RDM) remains underdeveloped. Thus, do not be surprised if you do not find what you are looking for.