GA pilot reported an NMAC during takeoff from BED airport requiring evasive action.

Narrative

I departed BED with my passengers onboard and a CFI friend on the right seat and myself as a PIC. The CFI; not giving any instruction on this specific flight; was unfamiliar with the aircraft as it was her first flight on one; she had no experience with the G1000 avionics either.I was instructed to line-up-and-wait on Runway 23 after Aircraft Y who performed a T&G. I was then cleared for takeoff; instructed to turn right 270 after takeoff and was advised the Aircraft Y ahead of me was doing left traffic.I took off and once at 800 feet I initiated my turn. I then noticed the Aircraft Y started turning right and the avionics started to complain about that traffic being close to us. In parallel the CFI in the right seat indicated she lost visual of that plane; but I still had it in sight. As I saw it turning right I disconnected the autopilot; stopped my right turn and decided to turn a bit to the left as not to cut in front of that traffic and pulled the controls as to tighten my climb. The Controller then inquired whether I had that traffic in sight; I indicated I did and commented I thought that traffic was doing left traffic which the controller agreed to and changed me to Approach after that. I ended up passing above that traffic by about 300-400 feet. I then re-engaged the autopilot; turned 270 and changed frequency.I spoke to the tower later that day; they indicated the Controller's belief the traffic ahead of me was assigned left traffic was a mistake. I'm not sure why the CFI in that plane didn't hear the Tower indicating that in their transmission to me (I believe it was a dual instruction flight).