The German education system

Most German children between the ages of 3 and 6 attend a German preschool, which is known as 'Kindergarten' or 'Kita' and they aren't run by the government, but by city or town administrations and local churches.
German elementary schools are called 'Grundeschule'. After children have completed four years of elementary school, there are four options for secondary schooling. The most gifted children attend a 'Gymnasium' (grammar school) and graduate after grade twelve or thirteen with an 'Abitur', an exit exam which qualifies you [them] for university. Then there is the 'Realschule', which students have to attend until grade ten. The least academic school is the 'Hauptschool' which students have to attend until grade nine and its exit exam is either a 'Hauptschulabschluss' or, after another year, a 'Realschulabschluss'. The last one is the 'Gesamtschule' (comprehensive school), which all students are allowed to attend, regardless of their school performance. Most German schools are public schools. A great portion of the German universities is [are] public, so students mostly don't have to pay tuition fees or [only pay] low ones. By the way, today's [today is] the Independence Day of Estonia!

The German education system

Most German children between the ages of* 3 and 6 attend a German preschool**, which is known as** 'Kindergarten' or 'Kita' and they aren't run by the government, but by city or town administrations and local churches.
German elementary schools are called 'Grundeschule'. After children have completed four years of elementary school, there are*** four options for secondary schooling. The most gifted children attend a 'Gymnasium' (grammar school) and graduate after grade twelve or thirteen with an 'Abitur', an exit exam which qualifies you for university. Then there is the 'Realschule', which students have to attend until grade ten. The least academic school is the 'Hauptschool', which students have to attend until grade nine and it's exit exam is either a 'Hauptschulabschluss' or, after another year, a 'Realschulabschluss'. The last one is the 'Gesamtschule' (comprehensive school), which all students are allowed to attend, regardless of their school performance. Most German schools are public schools. A great portion of the German universities is**** public, so student's students mostly don't have to pay tuition fees or low ones*****. By the way, today's the Independence Day of Estonia!

*It's fine, but this sounds better: "children aged between 3 and 6"

** The way you wrote it makes it sound like all the children go to a single school. Instead, I would write "German preschools". Also, another word for preschool, used more often, is "nursery".

** If you correct the previous line to "German preschools", then this one goes "which are known as". Another option is "also known as".

*** It's fine, but this would be better: "they have".

**** Should be "are". In English the verb corresponds to "German universities", not to "a great portion". It's a bit illogical, but that's how it works. Also, you could replace "portion" with "deal" to make it sound more fluent or a bit colloquial.

***** I suppose what you meant is: "...students mostly don't have to pay tuition fees or they pay low ones." You can't say it the way you did.

All in all, your English is pretty good! :)