As far as I know, Octavius had minimal contact with Caesar for any possible "tutelage" (C. Suetonius, De Vita XII Caesarum, Divus Augustus, Cp. VIII, sec. I-II):
According to Cassius Dio his relationship with Caesar were close:
"So much for Antony's conduct. Now Gaius Octavius Caepias, as the son of Caesar's niece, Attia, was named, came from Velitrae in the Volscian country; after being bereft of his father Octavius he was brought up in the house of his mother and her husband, Lucius Philippus, but on attaining maturity lived with Caesar. 2 For Caesar, being childless and basing great hopes upon him, loved and cherished him, intending to leave him as successor to his name, authority, and sovereignty." (Cassius Dio, 45.1)
And according to Nicolaus of Damescus Atia indeed object to her son joining thr military for fear for his life:
"Caesar had by this time completed the wars in Europe, had conquered Pompey in Macedonia, had taken Egypt, had returned from Syria and the Euxine Sea, and was intending to advance in to Libya in order to put down what was left of war over there; and Octavius wanted to take the field with him in order that he night gain experience in the practice of war. But when he found that his mother Atia was opposed he said nothing by way of argument but remained at home." (Nicolaus of Damescus, Life of Augustus, 6)