The alchemist helps Santiago in a variety of ways. Some of them are concrete and life saving, while other are more knowledge based. The first thing that the alchemist teaches Santiago is the importance of listening to his personal desires. Listen to your heart as the 80's singer Roxette would say/sing. Santiago listens to the advice and re-enters his "Personal Legend" of finding the treasure at the pyramids.
The alchemist also teaches Santiago about the Soul of the World. Through this teaching Santiago learns to read and understand the environment around him. He also learns to communicate with the world around him through some supernatural abilities. This leads to the concrete life saving I mentioned earlier. At one point Santiago and the alchemist are captured by Arab raiders. The alchemist tells the Arabs that Santiago will turn into the wind within three days. Spoiler alert, he mystically raises a big sandstorm and amazes the Arabs with his power, so they decide to let Santiago and the alchemist go.
In summary (and this is going to sound very cliche), the alchemist helped Santiago unlock his true potential.