which family to use in glm for non-binomial categorical variable

I'm working with a database to test seasonalities in different time frames. One of the response categories is binomial (rain and dry) and it's all right when I do the glm using the family="binomial", but the other categories are not binomial and I do not know which family to use to regress. One category is four factors (spring, fall, winter and summer) and the other is twelve (months of the year).