Yes, "to go to school" is the way (In the US, at least) we would normally refer to the school someone regularly attends. "She goes to private school," "He goes to Chinese school on the weekends," "She goes to a school with a really nice playground," etc. are all how we would normally discuss the school where someone is enrolled.
"Study" is usually for something more specific. It often refers to a particular subject or the act of physically sitting down and trying to learn stuff. "He's studying for the test tomorrow," "She's studying French this year, but thinking about switching to Turkish next year," and "It feels like all I ever do is study these stupid reflexive verbs," are all completely normal uses of "study."
It doesn't even have to be something associated with an actual academic school. I remember being 16 and studying for my driving test (It was very exciting!). Professional football players study their teams' play books. A singer might study the lyrics of a new song they had to learn.
"özel okul" means "private school" (so not public, you have to pay extra money to go there).
If we were talking about a school which is special in some way, then "special" would be used as an adjective and we would have to use "bir" in the Turkish sentence. "Kızım özel bir okula gidiyor"
As to my knowledge 'gidiyor' is the normal presence. If you say 'gider', what is the Aorıst tense, you are expressing that her habit is to go to school. But it is not her habit. She is actually visiting/attending that prıvate school. that is the reason why they use gidiyor; I think.