In book one, three notable acts take place.
First, Telemachus calls together the suitors, and he tells them to leave his father's estate. In short, he banishes them. This is a bold action, because Telemachus is still young, whereas the suitors are older men. In addition, the suitors have been present for a long time.
Second, when Telemachus sees that Penelope, his mother, is upset over the singing of a bard about the Trojan war, he tells her to leave. It almost comes off as a rebuke. This action shocks Penelope, because this action is out of Telemachus' character. Finally, Telemachus says that he will take care of the suitors.
Antinous and Eurymachus, two unruly and defiant suitors, give Telemachus trouble, but Telemachus is undeterred.
Third, Telemachus sets a course to Pylos and Sparta to gain more information on his father.
These three points show that Telemachus is growing up. The catalyst in all of this is the visit of Athena in the disguise of Mentes, a friend of Odysseus. Without the encouragement of Athena, none of this would have happened.