One major theme is pride vs. humility.
Bree is a very proud horse. He knows he is smarter than the non-talking horses of Calormen. He knows he is from Narnia, and is proud of it. But he is also proud of many other things, such as being an experienced war horse, of the important people he knows in Calormen, and of his appearance.
However, like many proud people, Bree doesn't realize how much he doesn't know. He actually knows very little about Narnia. And though he believes in Aslan, he thinks it would be silly, undignified, if Aslan were literally a lion. This leads to a very funny scene where Bree is smugly declaiming about how Aslan can't really be a lion, only to jump and scream in terror as Aslan's whiskers tickle him from behind.
In many ways, The Horse and His Boy is the story of the progressive humbling that Bree undergoes.
Hwin, on the other hand, is humble and sweet, and this prepares her well to meet Aslan.
Similarly, Aravis is proud. She is a Calormen aristocrat, and though running away from her homeland, she is still a snob. She looks down on Shasta for his lower-class ways. Little does she know that he is really a prince (he doesn't know it either). Aravis is gravely humbled when the lion rakes her back with its claws, and Shasta runs back on foot to save her. Then she is humbled again when she finds out that Shasta is now the heir to the throne of Archenland.
Through Bree, we see how ridiculous pride can make us. Through Aravis, we see how it can cause us to undervalue others. Through Hwin, we see how Aslan loves and values the humble.
I would say that one of the main themes is independence and indpendent thinking. Shasta, for example, spends much of the book learning to think for himself and "shedding his slave mentality", and learning how to both think and act for himself. On the other side of the coin, Bree also wants to be "free" but he has a different path to follow.