Cessna 180 pilot reported being close proximity to another aircraft during the initial climb out. Pilot remained in pattern and landed.

Narrative

I completed my run up in preparation for a takeoff from XX. I and my passenger did not hear any radio transmissions during that time and performed an S turn to look for traffic. I did not see or hear any traffic on final. I announced that I was taking XX and intended to stay in the pattern before lining up on the runway and doing a final run up check before departure. Shortly after takeoff; I heard a broken radio call that indicated we had cut someone off on final. I continued on course while trying to determine where the other aircraft was at. I rocked my wings; a little to provide better visibility and heard a second radio transmission that said we are on your left side; Don't come any closer. I spotted the Aircraft shortly after that; off our left wing tip and slightly higher than our altitude. I continued my climb out while maintaining visual reference on the other airplane. I completed about four circuits in the pattern behind the other aircraft and observed multiple approaches that were significantly low and very far off of center line. I also observed the other Aircraft making an extended downwind leg and calling final approximately 4 miles from the airport with no other calls until turning crosswind. Towards the end of my time in the pattern the traffic increased to approximately four student aircraft; myself; a Cirrus and a Leer jet. I observed multiple radio calls being stepped on by other aircraft in the pattern without any follow up calls. I think the unorthodox approach and radio calls led to my failure to see and avoid that aircraft before takeoff. Upon our final landing and taxi back to the hanger; I stopped at the flight school and attempted to talk to the CFI about what happened. My primary goal of the conversation was to get their perspective on what happened so I could properly determine a solution for avoiding this in the future. They were unwilling to talk about the situation other than to say that I was dangerous and should be ashamed of myself. I feel like it was not in the best interest of figuring out what happened and trying to improve safety. In the future; I will take into account that with increased student traffic at our airport. There may be times when airplanes are not in the place they are expected to be. I intend to do a full 360 before taking the runway to aid in looking for other traffic. I will also do a radio check to make sure that I am transmitting and receiving calls properly.