These days I have been making my first money by tutoring. I also changed my workplace/place of work so I am genuinely busy. I hope I can keep up with the pace of the work.

These days I have, for the first time, been making money by tutoring. Since I changed my work place, I have been genuinely busy. I hope I can keep up with the pace of this new work. There is no logical implication between changing your work place and being busy, so "so" is not the best word choice. What you want to say is that you have been busy SINCE the time when you changed your work place.

I have recently started earning money by tutoring. I also changed my job, so I am genuinely busy. I hope I can keep up with the pace of the work.

These days I have been making extra money by tutoring for the first time. I also changed my workplace, so I'm genuinely busy. I hope I can keep up with the pace.