From the point of view of the books, Arwen really is an off-scene character throughout. There is a back story there that is only barely touched upon in the Lord of the Rings with the mentions of her throughout, as you noted. Being very young from an Elven point of view as well as being of the equivalent of royal blood, she is largely protected and alternates between Rivendell (where she first met Aragorn) and Lothlorien. Aragorn and Arwen's love affair is only briefly sketched out, so within the immediate story she only really exists as a paramour of Aragorn. Think of her as a royal princess - she's not about to go out hunting Orcs or doing much outside her protected realms.
The movies beefed up her role a fair bit, with her supplanting Glorfindel in carrying Frodo to Rivendell, some scenes in Rivendell with Aragorn, and her coming with the Dunedain to Aragorn's aid outside the Valley of the Dead, before returning home. This seemed to be Jackson's attempt to provide more of a feminine touch to at least parts of the film, as well as an attempt to address why Aragorn would choose Arwen over Eowyn - something people who came to the movies without reading the book reportedly questioned.