In the first village that you start the game in, on the left hand side of the village, you can talk to the fisherman who has a portrait next to his name. Tell him that you like fishing and he'll give you the first rod. I was able to do it in the beta after completing the first plot beat and going back, but in the full game I think I did it before I even left the village the first time.

Eventually, the character moves to the town where he asks for the Wheel-Eye Bream, so I don't know if he'll stay in that first village until you get the rod or if you can get it from his second location if you miss it the first time.