If you are visiting the bigger cities (St. Louis or Kansas City), then most people will pronouce it as "MissourEE". If you are in the outstate portion, especially in the southern part of the state, they all pronounce it "MissourAH" or "MissourUH".
And for God's sake, if you are in St. Louis, do NOT call it "Saint LouiEE". Nobody says that and if you do they will think your are a total ***! It's "Saint LouIS."
As a Life long resident of Columbia,as a son of a Columbia School Teacher for over 30 years.Mgard27 post is incorrect ....Sorry but a professor at the University of Columbia where both my mother and brother graduated anounced some years back that both ways of saying missouri is correct both.. miss-oree or miss-ori is correct.i.At different times i use both versions.Thought the visitors might like some advice from a Real Missourian!!