Was it really the Coach's first season with the team?

No. As stated above, it was Coach Wood's second season with the Milan Indians. He actually took them to the semi-finals the previous year. During his second year, which is what the film depicts, the town was no longer skeptical of his new strategies for offense and defense. The town was behind him. This is opposite to what is shown in the movie, where a community referendum is held to determine the Coach's fate.