In T. H. White's The Once and Future King, what are some quotes from Book I that demostrate Wart is using what he has learned?
The best passages in Book I that demonstrate Wart putting into practice all that Merlin has taught him are found when he is either about, or already attempting, to pull out the sword.
For instance, when Wart walks into the churchyard, he notices that he sees "everything much more clearly" (Ch. 23, p. 203). He sees that the gargoyles look beautiful, he sees the striking colors of the banners as they wave, he sees that the snow looks clean, and notices scents. The increase of his perceptions are due to Merlin's training. Wart's interactions with the animals in their foreign environments taught him to see things, smell things, and sense things that he would not have otherwise. The increase of his perceptions will help with his role as King, because his senses will guide him in his decision-making process.
Another good passage showing Wart applying what he has learned is the passage describing Wart pulling out the sword. All of the animals he has met through his education come to coach him in his endeavor. Luce, the pike, reminds Wart to "put [his] back into it" and that "power springs from the nape of the neck." Not only is Wart's lesson in agility useful for empowering him to pull out the sword and claim his birthright, his lesson will be valuable for his duties as a warrior, which is required of kings.
Similarly, in this scene Merlin in the form of an owl reminds him, "What is the first law of the foot? I thought I once heard something about never letting go?" Merlin reminds us that Wart's lesson as a falcon has taught him the value of persistence, which he is not only applying now, but will also apply in the future as King.