Gladly! Here are my corrections:
I started my day with an italki class with teacher Raymond. After that I went to work. At about noon, I took my kids to school, had lunch with them, and then returned to work again. I worked until 6 pm and then went to the gym. Then I went home.
The prompt is asking what happened today, but I noticed you are describing what you generally do each day. If that's what you would like to express, you would say it like so:
I usually start with my day with an italki class with Raymond. After that, I go to work. At noon I bring my kids to school where we have lunch before I return to work. I work until about 6 pm and then I go to the gym before finally returning home for the evening.
And lastly to answer your question, we say "to school". I take my kids to school. I go to school Monday, Wednesday, and Friday, etc.
Let me know if you have any other questions!