I don't know if it was over heating prior to the water pump replacement

Previous owner was driving it and heard the water pump squealing. Finished driving home roughly 5 miles. Parked it and replaced the water pump. I don't know if it was over heating prior to the water pump replacement or after. Any ways I started it up and it heated up pretty quick within a mile or so. The temp gauge went to redline and the warning buzzer dinged. I had turned on the heater right before it dinged. I felt the air and it would alternate from warm to cool air going down the road. The temp did lower a bit about 210 and then went back up and dinged again but quickly went back down and fluctuated from there on and never got back up where the warning would ding. But still would alternate from warm and cool air. Never did get hot air. When I parked it I could hear and see it periodically steaming out the side of the water pump where the steel tube is seated on the drivers side of the water pump. What is the possibilities for this.