-The making of the Fellowship
In the books, Elrond chooses the companions for Frodo. “For man you shall have Aragorn son of Arathorn, for the Ring of Isildur concerns him closely.” (FotR, The Ring goes South) After that, Aragorn again asks Frodo for leave to be his travelling companion.

In the films, Aragorn is the first to volunteer his services, “If, by my life or death, I can protect you, I will. You have my sword!” (Council of Elrond) This is almost exactly the same wording as Aragorn’s earlier vow to Frodo in the book.-

For correct sources, this is the excerpt from the script of FotR on this website: Frodo: I will take it! I will take it!
Frodo: I will take the Ring to Mordor. Though.. I do not know the way.
Gandalf: (walks towards Frodo) I will help you bear this burden, Frodo Baggins, so long as it is yours to bear. (puts his hand on Frodo’s shoulder)
Aragorn: (rises) If by my life or death, I can protect you, I will.
(kneels before Frodo)
Aragorn: You have my sword.
So, just so everyone knows, Aragorn was NOT the first to join Frodo in the Fellowship, it was Gandalf…no harm done. 🙂