Samuel is not not happy with this request—the Israelites are not supposed to be like the other nations, they’re supposed to be set apart to the LORD and He is supposed to be their leader—but God tells Samuel to go ahead and submit to their request. God then appoints Saul to be the Israelites’ first king.
At first, Saul seems to be a good choice. However, it soon turns out that he is willing to take shortcuts and disobey the LORD when he deems it convenient, so God removes Saul as king, and sends Samuel to anoint David as Israel’s new king.
For those who don’t know the story, one might assume that David would then take over as king, but this is not what happens. Instead, although the LORD has removed His blessing from Saul, He leaves Saul as king for quite a number of years after. He even institutes David as Saul’s servant, where David plays music for Saul when Saul is having one of his mental episodes! Eventually, however, Saul becomes convinced that David is going to usurp his throne, even though David has never made a move to do so—in fact, David absolutely refuses to do so on numerous occasions, refusing to lift a hand against “the LORD’s anointed.”
Eventually, David has to go into hiding, because he fears for his life, what with Saul wanting to kill him. Once this happens, the rest of I Samuel seems like a downhill journey for Saul, as we simply wait for his kingship to be over, and an uphill journey for David, as we see him maturing during his time in hiding. The book ends with Saul’s death, but without David taking over in his stead; for that, we’ll have to wait for II Samuel.