C172 pilot reported a NMAC during departure climb and took evasive action to avoid a collision.

Narrative

At XA:45; I departed ZZZ airport for ZZZ1 airport; flying VFR. I was sitting in the right seat and was Pilot In Command. Another pilot was sitting in the right seat and was the sole manipulator of the flight controls. Both of us have a PPL (Private Pilot License) and are current. I am also instrument rated and current. While holding short of Runway XX; we noticed an aircraft flying nearly directly overhead to the east. I called on the CTAF and asked that aircraft to state their intentions; as they were not flying in a normal established traffic pattern. Runway XX is right traffic; and they were just south of the runway; essentially overflying the taxiway. The pilot of the other aircraft stated that they intended to land on Runway XX; and started to maneuver towards an approach. I judged that we had plenty of time to depart; so we called on the CTAF stating that we were taking Runway XX for a departure; and were following the local noise abatement procedure; which prescribes a 30 degree turn to the north after take-off. We departed and made the 30 degree turn to the north to avoid overflying the houses on the west end of the airfield; and began a climb to 6;500 ft. MSL. Once we were 2 miles west of the field; I called on the CTAF and stated that we were departing the area to the west and that this would be our final call on that frequency. There was no answer after about 10 seconds; and I switched to ZZZ Center and requested flight following to ZZZ1. After issuing us a squawk code; the Center Controller warned us that there were two targets nearby; one target about 1;500 ft. above us which was no factor; and a second target; an Aircraft Y which was at the same altitude; just south of us; also climbing. I told ATC that we had the traffic in sight and may maneuver to avoid. A moment later the Aircraft Y turned directly toward our flight path. I told the pilot in the left seat to stop our climb so we could gain airspeed and pass below and in front of the climbing Aircraft Y. I believe the other airplane passed above us no less than 200 ft.; and behind us no less than 300 ft. The flight continued to ZZZ1 uneventfully.