Círdan may have been among the Elves who awoke at Cuiviénen, although Tolkien never explicitly states this anywhere. When the three chiefs (Ingwë, Finwë and Elwë) returned from Valinor, they told their people to come from Cuiviénen and to go there.
Many groups of Elves broke away and formed the Nandorin and Silvan Elves. Elwë, Olwë and many of their people (the Teleri) continued on their journey, with Círdan among them. Twenty years after they first left Cuiviénen, they entered Beleriand. They were behind the Vanyar and Noldor by a long distance. On their way to the western shores, they passed through Nan Elmoth, where Elwë and Melian met and fell in love. After Elwë's disappearance, his brother Olwë led the remaining Teleri.
Círdan was with Olwë when they came to the shores of Beleriand, where the Teleri built ships, for the Noldor and the Vanyar had left three years earlier with the help of the Vala Ulmo. Ossë and Uinen befriended the Teleri when they reached the shores and helped them as well as they could. After the ships were built, in a final attempt to find Elwë, who had been lost to them since he had entered the woods of Doriath, Círdan and a small band of elves sought him once more. Many Teleri chose to live on the shores of Beleriand, rather than cross the sea. Yet Círdan was insistent on travelling to Aman, even if he had to go alone. Some few chose to go with him, and he set about the task of building a vessel to bring them there. On the night before their departure, Círdan had a dream in which he received a message from the Valar.
"And the voice warned him not to attempt this peril; for his strength and skill would not be able to build any ship able to dare the winds and waves of the Great Sea for many long years yet. "Abide now that time, for when it comes then will your work be of utmost worth, and it will be remembered in song for many ages after." "I obey," Círdan answered, and then it seemed to him that he saw (in a vision maybe) a shape like a white boat, shining above him, that sailed west through the air, and as it dwindled in the distance it looked like a star of so great a brilliance that it cast a shadow of Círdan upon the strand where he stood." The History of Middle-earth, vol. XII, The Peoples of Middle-earth: "Last Writings - Círdan," p. 386