The circumstances in chapter 13 indicate to me that perhaps I was too hasty in my evaluation of chapter 7. Let's see.
Saul has managed by this time, about two years on, to amass a regular army (sort of, we'll see some oddities later in the chapter) and has put a division under the control of his son Jonathan, who attacks the Philistine garrison in Geba. The Philistines gather their forces to retaliate, and Saul amasses the Israelite fighters at Gilgal, waiting for Samuel to come and bless them before they go out to battle.
The situation looks bleak, and it seems that some of the men are deserting. Samuel is late in showing up, and Saul starts to get worried. So Saul decides to go ahead and start things without Samuel: he burns a sacrifice to God. As was suggested a few chapters back, it may have been wrong for Saul as a non-Levite to offer a sacrifice. Whether or not that was the problem here, it seems clear that he was not supposed to perform this action, and he gets in big trouble for it.
Samuel arrives, and he knows that Saul has done something wrong. Saul admits that he was afraid to go to battle without having given a sacrifice, and since Samuel was late in coming... Samuel won't have it, though, and tells Saul that God is going to give the kingdom to someone else.
Yes, that person was David, who was not exactly the paragon of virtue, but I'll reserve the bulk of comments on David's character to the chapters in which David actually appears. For now, let it suffice to say that Saul is losing the kingdom because he's not following the religious rules, which to God are extremely important. David expresses in quite a bit of his poetry that he would rather be a priest than a king, but never acts in a way that indicates he doesn't know his place in the end.
In the end of this story, we're told that while there are a lot of men ready to go to battle against the Philistines, there are at this point in history (generally) no swords or spears in Israel; as a bunch of farmers, they're going to war with sharpened farming tools.