These two books tell of Samuel, Saul, David, and Solomon; and the beginning of the kingship of Israel. It was not a peaceful process! Samuel was the son of Hannah, who was previously unable to conceive. Happy to be blessed with a child, she gave Samuel to serve God, and Samuel became a prophet.
What I noted in reading this is the tendency, even then, for people to want to "fit in" and be like everyone else.
1Sa 8:4-9 Then all the elders of Israel gathered themselves together, and came to Samuel unto Ramah, And said unto him, Behold, thou art old, and thy sons walk not in thy ways: now make us a king to judge us like all the nations.
But the thing displeased Samuel, when they said, Give us a king to judge us. And Samuel prayed unto the LORD. And the LORD said unto Samuel, Hearken unto the voice of the people in all that they say unto thee: for they have not rejected thee, but they have rejected me, that I should not reign over them. According to all the works which they have done since the day that I brought them up out of Egypt even unto this day, wherewith they have forsaken me, and served other gods, so do they also unto thee.
Now therefore hearken unto their voice: howbeit yet protest solemnly unto them, and shew them the manner of the king that shall reign over them.
They did not trust God, but wanted a king to rule over them like the other nations.
Samuel did as God instructed and told them what a king would do for them and with them (little details like taxes), and then anointed Saul at God's command as the king. However, Saul did not stay true to God either and was not obedient. First, he did not wait for Samuel before a battle as instructed, but instead made an offering himself. It seemed like the right thing to do to him, but was not obedient to God's plan.
Later, Saul did not carry out the instructions of God regarding the destruction of the Amalekites. These events cost Saul the kingship, though he remained in power until his death when David was appointed king.
1Sa 15:9-23 But Saul and the people spared Agag, and the best of the sheep, and of the oxen, and of the fatlings, and the lambs, and all that was good, and would not utterly destroy them: but every thing that was vile and refuse, that they destroyed utterly.
Then came the word of the LORD unto Samuel, saying, It repenteth me that I have set up Saul to be king: for he is turned back from following me, and hath not performed my commandments. And it grieved Samuel; and he cried unto the LORD all night.
And when Samuel rose early to meet Saul in the morning, it was told Samuel, saying, Saul came to Carmel, and, behold, he set him up a place, and is gone about, and passed on, and gone down to Gilgal. And Samuel came to Saul: and Saul said unto him, Blessed be thou of the LORD: I have performed the commandment of the LORD.
And Samuel said, What meaneth then this bleating of the sheep in mine ears, and the lowing of the oxen which I hear?
And Saul said, They have brought them from the Amalekites: for the people spared the best of the sheep and of the oxen, to sacrifice unto the LORD thy God; and the rest we have utterly destroyed.
Then Samuel said unto Saul, Stay, and I will tell thee what the LORD hath said to me this night. And he said unto him, Say on. And Samuel said, When thou wast little in thine own sight, wast thou not made the head of the tribes of Israel, and the LORD anointed thee king over Israel? And the LORD sent thee on a journey, and said, Go and utterly destroy the sinners the Amalekites, and fight against them until they be consumed. Wherefore then didst thou not obey the voice of the LORD, but didst fly upon the spoil, and didst evil in the sight of the LORD?
And Saul said unto Samuel, Yea, I have obeyed the voice of the LORD, and have gone the way which the LORD sent me, and have brought Agag the king of Amalek, and have utterly destroyed the Amalekites. But the people took of the spoil, sheep and oxen, the chief of the things which should have been utterly destroyed, to sacrifice unto the LORD thy God in Gilgal.
And Samuel said, Hath the LORD as great delight in burnt offerings and sacrifices, as in obeying the voice of the LORD? Behold, to obey is better than sacrifice, and to hearken than the fat of rams. For rebellion is as the sin of witchcraft, and stubbornness is as iniquity and idolatry. Because thou hast rejected the word of the LORD, he hath also rejected thee from being king.
Meanwhile, David works for Saul, slays Goliath, and in general kicks some butt with the army. Saul becomes jealous and attempts to kill David several times, and David ends up fleeing. Saul dies in battle, and David is crowned King. This is a very aggressive summary of events - this takes about a book and a half in the Bible and moves us into Kings, which is the next book.