Correction:     Student's sentence        Corrected sentence        Grammar explanation       Better sentence   

My Dream is to be a teacher in kindergarten and daycare center.

My dream is to be a teacher in kindergarten and in daycare center.

"Dream - dream" - common noun

So, I'm majoring early childhood education at the university. 

With this, I'm majoring early childhood education at a university. 

"the - a" - the can also be correct but the in not that specific

My hope is that the children I educated will grow up well and have a good social life.

My hope is that the children I educate now will grow well and will have a good social life.

educate - present tense 

I also want to study my current major hard and become a good teacher!

I also want to study well about my current major and become a great teacher someday!

"hard - well" - adverb