Despite the fact that Through the Looking-Glass is structured around a chess game, there isn't really an antagonist for Alice to struggle against. You'd think that the Red Queen, the most powerful piece on the opposing side of the game, would be the antagonist. However, the Red Queen actually assigns Alice her role in the game and gives her advice on how to succeed. When Alice makes it to the eighth square and becomes a Queen in her own right, the Red and White Queens are both there to celebrate with her. The conflict between the red and white pieces is only a structure, not a bitter battle, and the Red Queen is more of a guide and mentor than an antagonist.
The Caterpillar is the first character in Alice's Adventures in Wonderland who gives Alice straightforward advice on the problems she's been having and a concrete solution. The Caterpillar listens to Alice's story of constantly changing sizes and mis-reciting poetry and explains that she can control her size by eating different bits of the mushroom. Of course, at the time, Alice thinks the Caterpillar is extremely rude. But then, we don't always appreciate being guided and mentored, do we?
The Cheshire Cat is one of many characters who guide Alice on the various stages of her wandering journey through Wonderland. Although the Cat doesn't give any useful advice (the best it does is to explain that it doesn't matter which way Alice goes because everywhere leads somewhere), it does listen to her and functions as an ally. At the home of the angry Duchess and during the strange game of croquet, the Cat seems to have a special bond with Alice. The Cat confirms for Alice that the behavior of everyone around her is bizarre and helps reestablish her sense of reality. (Well, except for the appearing and disappearing part.)
The Gryphon isn't one of the more famous characters in Alice's Adventures in Wonderland, probably because he was left out of Disney's cartoon version, but for two chapters he's Alice's friend and advisor. In fact, he's a more significant character than the participants in the Mad Tea Party, who get so much attention. The Gryphon is the one who takes Alice away from the crazy croquet game, who introduces her to the Mock Turtle, and who helps her try to understand how her adventures have changed her. The Gryphon also lets her in on a secret about the Queen of Hearts's court: they don't really execute anyone.
Humpty Dumpty acts like a teacher for Alice, explaining the strange poem she read in the book in Looking-Glass House line by line. He also teaches her about the flexibility of language and how easy it is to twist words around – as long as you don't mind nonsense.
The White Knight is one of the many characters in the Alice books who seems to represent Lewis Carroll himself. The Knight doesn't teach Alice anything intellectual – in fact, all his ideas and inventions seem foolish. But he does accompany her through the last few squares of the chess game, protecting her from the Red Knight and acting as a protector and companion.