C172 pilot reported they turned the wrong direction on initial climb and flew below the minimum vectoring altitude.

Narrative

Awaiting a VFR on top clearance at ZZZ Airport; I received clearance via Flight Service; and they cleared me:Cleared ZZZ VOR via ZZZ ODP (Obstacle Departure Procedure); direct ZZZ VOR. Climb 4;000 ft; expect 5;000 in 10 mins. Clearance void if not off by 3 mins.Clearance asked me if I was ready for takeoff and I said affirmative. ZZZ is an airport which I am familiar with in VFR conditions; but this was my first Instrument Departure without a CFI-I on board. I announced my intentions on the CTAF; and took off runway XX; I switched to ZZZ Approach on frequency. I turned left crosswind; still under the cloud layer; when ZZZ approach wanted to confirm that I had turned right; and not left. I explained that I had mistakenly turned left. They asked me if I had the terrain in sight; and by now I was on left downwind; climbing through the cloud layer. I said that I had the terrain in sight right before I turned left downwind; and the terrain to my right was not a factor; as it was adjacent to me. I broke through the clouds; and I was headed Eastbound; direct ZZZ VOR. ZZZ approach said that they had a number for me to copy for a possible pilot deviation; and I copied it and called them later after I landed. I cancelled IFR; since it was a VFR on top clearance; and I continued flying to ZZZ1 airport as planned for a short cross-country.I understand that this deviation was no one's fault but my own. It was a result of my inexperience of receiving IFR clearance in this airport and forgetting to look at the Obstacle Departure Procedure prior to takeoff; even if that meant needing to get a new clearance if it took longer than 3 minutes to get myself composed for the departure. I understand the danger that I put myself in; especially if this were another airport and I was not familiar with the terrain; and if Approach did not query my deviation. I believe my familiarity with the airport environment was a contributing factor to my complacency; since I was familiar with the local terrain. I forgot to check the ODP; even though Clearance told me to follow it. I believe that because of this experience; it will be unlikely for me to make this same mistake in the future; whether it is here or at any other airport. This situation can be avoided in the future by taking my time and double checking that each step of the clearance is understood and reviewed before taking the active runway.