Can I use 告诉 to quote what someone has said? Please help!
今年初，我的朋友告诉我他这个学期每个星期要打工三十个小时。
It is correct. You can use 告诉 in this way.

You can add a comma to make it clearer.
今年初，我的朋友告诉我，他这个学期每个星期要打工三十个小时。  

BTW, the sentence can be simplified as follows.
今年初，朋友告诉我，他这个学期每周要打工三十个小时。
今年初，我朋友说，他这个学期每周要打工三十个小时。