A C172 pilot reported they deviated off course from the SID and received a Low Altitude Alert from ATC.

Narrative

flying ZZZZZ departure off of runway XX ZZZ. Pilot error; unfamiliar with the airspace and was confused about having to make a climbing right turn instead of simply turning left toward the ZZZ VOR. I was supposed to make a right turn but made a left turn just after being handed off to departure. ZZZ TRACON immediately asked me to confirm I was in a right turn; I said negative; I'm turning left; but now I'm reversing course and going right. ATC then issued a low altitude alert as I mistakenly turned into a sector with a minimum altitude higher then I was at; and asked me to confirm that I can visually avoid terrain; to which I said affirmative. I was then given heading vectors for the rest of my climb until I was cleared back onto my planned route. I apologized for my mistake; and he explained that this happens relatively often and usually when it does; departures out of ZZZ1 have to be stopped; but since I corrected my mistake so quickly that was not the case this time; so no big deal after all. Contributing factors include fatigue and confusion; this happened at the beginning of the final leg of a two day; 22 hour journey. Thankfully as mentioned above the problem was quickly caught and corrected.