The bad and abusive one was the Baltic-Swedish tutor Otto Friedrich von Brümmer (engaged by Peter III's uncle, after his father's death, dismissed by his aunt in Russia), but the tutors chosen by his father, who remained as assistants during Brümmer's time, were good tutors: Pastor Gustav Christoph Hosmann and the French teacher Monsieur Mild. In addition his aunt Elisabeth engaged a very good German tutor for him in Russia: Professor Jakob Stählin.
He was also accompanied on his journey to Russia by a very good and caring scientist: Johann Albrecht Korf. He wrote an account of the journey and portrayed the young Peter as an intelligent and sensitive youngster. He had to leave his beloved dog in Kiel and was rather upset about this. (Remember he was only 11-12 years old and had lost both his parents. The dog was, together with his tutors and his trusted and loyal equerry / valet Bastian, his friends and family.) So one of the things he was most curious about regarding Russia was Russian dogs, dog breeds and if he could get a new dog in Russia.
His father was a rather educated man and ruler and took great care of his son's education untill he died. His aunt Empress Elisabeth, although uneducated and frivolous herself, took a wise and careful approach to her heir's education, at least in theory. (One problem was the many interruptions in the schedule due to the many parties and balls she hosted and wanted Peter to attend.) All this according to Elena Palmer.