As a starting point, you may take a look at the PISA study. It is conducted by the OECD in ca. 80 countries around the world. It uses various indicators to assess (1) reading performance, (2) mathematics performance, and (3) science performance on a country-level (disaggregated by gender and age group).
Here is an example visualizing the 2018 mean reading performance of girls and boys, with the x-axis designating various countries (source):
For more detailed information on the PISA study and its methodology, I would suggest you to browse through OECD's library, or to directly go to the PISA website.
As regards higher education institutes, the indicators will differ significantly - usually, they deal with research outputs in scientific journals, the number of citations accrued by those publications, the share of national and international collaboration with other researchers around the world, the amount of grants and funding obtained by the institute's faculty, or the teaching-staff-to-student-ratio. One illustrative example is the THE World University Ranking. (But note that the reliability and credibility of these rankings may be subject to biases and methodical flaws.)