From tomorrow onwards means starting tomorrow, and the days afterwards. For instance: I used to drive to school. From tomorrow onwards, I'll go by train. "From tomorrow on" would not work there, but you could just say: From tomorrow I'll go by train. From tomorrow I'll take the train. Starting tomorrow I'll go by train. These sound more natural in everyday English. "From tomorrow onwards" is absolutely correct, but it's more something that someone like a professor would say.