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S/he will start to school on next year.

He or She will start to the school next year.

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Yes. We say both of them:

1)Okula başlayacak.

2)Okula gitmeye başlayacak.

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To start something means  "bir şeye başlamak or bir şey yapmaya başlamak"

Okula başlamak to start to school

Start an engine means "motoru çalışır" in Turkish. I dont know why did you say person can not start ?  

He will start to (the) school. Okula başlayacak.

She will start to play piano. Piyano çalmaya başlayacak.

I will start to work. Çalışmaya başlayacağım.

I will start to study. Ders çalışmaya başlayacağım.

So, people can start to do something.

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Only bilmek and istemek can take the basic form of the infinitive. In other verb chains the infinitive takes the form which any noun would take in that place. Like başlamak + DAT.

Okula gitmeye başlayacak.

-e başlamak. this verb always needs this suffix before it."he starts going", so we add "-a" suffix to the verb. you add the -e suffix after the thing you start, it can be either a verb or a noun. 

okula gitmek. gitmek also needs a "-e" suffix. so there are two suffixes belonging to two verbs in this sentence.

-i istemek. İstemek takes a -i suffix.so, okula gitmeyi istiyorum.

edit: you can say "okula gitmek istiyorum" as well. it is a more general form and indefinitive.