The lesson of loyalty is one of those lessons that all readers can take from White's story. Charlotte is loyal to Wilbur and Wilbur to Charlotte. This is a dominant characteristic of both throughout the novel and is one that can be taken away as a lesson. The idea of being able to stand for and by someone is extremely important, and one worthy of learning. In a larger sense, there is also something shown in the novel about the nature of animals, which can be taken away from it. The development of a consciousness about the lives led by animals and the behaviors they show towards one another brings a level of compassion within the reader. To develop this consciousness towards animals, especially at an early age, is something that can constitute as a lesson learned from the work.
One thing that I think you can learn from this book is that true friends are very important. If it were not for Charlotte, Wilbur would have gotten killed. But Charlotte goes to the trouble of writing all these words in her web to save his bacon (oooh, that was a funny one).
I think another thing that you can learn from this is that you should not prejudge how valuable something can be. At first, Wilbur seems like a runt and is going to get killed. Even so, he ends up winning a prize at the fair.