Lessons focus on c3 because it is a more forcing variation, as White is potentially intending to play d4 very quickly and for this reason it is important you respond ...Nf6, no other move will do.
Moves like O-O and d3 are less challenging, but you can still play Nf6, it's just that other options like d6 are equally good, so it really doesn't matter what you play as long as it's a reasonable move. Openings like the Italian get too nebulous for concrete theory quickly in many lines, which is why masters like playing them.
I am a beginner (ELO 1000) trying to improve how I play the Italian game when I am black. The lessons I have found all seem to assume that after
1. e4e52. Nf3Nc63. Bc4Bc5
white plays 4. c3.
In the 30 odd games I have played this opening as black, white has only played 4. c3 twice. Usually, it is d3 or 0-0.
What is the best response?