Why was God angry at the Israelites for choosing Saul as king?
This is a good question, because in Deuteronomy 17 God allows for the people to have a king. So, the anger of God is not based on the fact they wanted a king, but the reason why they wanted a king. Let me explain.
First, the historical context is very important. 1 Samuel, the book which speaks of the Israelites desire for a king, comes right after the book of Judges. The time of the book of Judges was one of the darkest times in Israel's history. The book actually ends with a civil war. Part of the reason for this chaos was the fact that the Israelites wanted to be like the other nations. What they needed instead was a king who would lead them to keep the covenant.
Second, the book of I Samuel is also in the period of the Judges. In fact, Samuel is the last judge. The people in this book wanted a king, so that they could be like the other nations. This shows that they did not learn much at all! Why? They must not be like the other nations. They were actually called to be different.
"Behold, you have grown old, and your sons do not walk in your ways. Now appoint a king for us to judge us like all the nations."
1 Samuel 8:5 says it all.
This is precisely why God gives them Saul. He was handsome and tall. According to 1 Samuel he was a head taller than anyone else. In other words, he was just like the kings of the other nations. What the Israelites should have asked for was a king who was godly, like David. This is certainly the thrust of the theology of 1 Samuel, because when God anoints David as King, he reminds Samuel that God is not like man, because he looks at the heart. 1 Samuel 16.
Firstly, the children of Israel did not choose Saul. They accepted him as God's choice for his people. But to answer your question God was wroth because they were asking to worship a man, who could draw them away from following him wholly. This eventually happenned after the passing away of Solomon the third king of Israel.