A good example of this is Onfim, a 13th Century Russian boy whose writings were discovered in the mid-20th century.

A brief overview of his writings can be found here: http://www.goldschp.net/SIG/onfim/onfim.html

A child in Middle Ages Europe may or may not have had schooling, depending on the time period, locality etc.

As you can imagine, if you were born the son of serfs in Europe, such as the Holy Roman Empire, you would most likely be working the land by age 5 or 6 alongside your parents, learning the farming trade.

Conversely, if you were in a larger city and the son of a merchant, artisan, etc. you would have more opportunities for education - which was overseen by your local church. You would learn, primarily, the basics of reading and writing, basic math for use in money handling and counting/division, and this was primarily supplemented by the local language version of the Bible - which in the early Middle Ages was usually in Latin, and later translated to the various languages such as old French, Middle German, Aragonese, and Church Slavonic in the Eastern regions.

If you check Onfim's writings you can see that the son of a city guard/soldier ( he refers to his father as a warrior in the drawings ) at least had basic writing education.

After basic education, you'd either be taught by your parents their trades/occupation or whomever they could secure an apprenticeship with.

Nicholas Orme's Medieval Childhood (2001) is a wonderfully illustrated and detailed survey of what children did from birth to adolescence, with a heavy emphasis on English children. It covers work, play, education, etc. It's a scholarly book but is easily intelligible to general readers.