Student Pilot reported a NMAC while on initial climb out.

Narrative

The aircraft I was in was on an IFR flight from ZZZ and going to ZZZ1. I was on ZZZ Tower XXX.X. Tower instructs me to line up and wait on Runway XXR. There was departing traffic on the upwind of XXR. I was getting ready for the takeoff; and I kept in mind of the departing traffic. I lost sight of them when Tower instructed me; 'Aircraft X clear for takeoff Runway XXR fly runway heading'. As I was on a takeoff roll; the gauges were green; and the airspeed was alive. I rotated and flew runway heading. I was between 625 to 825 MSL when my G1000 system gave me an audible warning; 'Traffic; 100 feet below.'. The one thing I did not know was if there was right traffic for XXR or if it was left traffic. I kept flying straight out and increased my climb to VX to get more distance. As I was climbing I saw the aircraft; which was a Cherokee turning left some 100 ft. below me. ZZZ Tower did not give me any traffic alerts. They were very busy; and I was trying to get a phone number from them to do reporting but could not get a chance to. I kept flying; and ZZZ Tower told me to switch to Approach. I got radar vectors and continued on with my flight plan.