Her academic accomplishments resulted in degrees in neurobiology, plasma dynamics, and eco-regeneration (with honors). She was accepted to Starfleet Academy and selected out of hundreds of applicants for an internship on the USS Enterprise-D.
In 2369, she traveled to Starbase 112 where she transferred to the Enterprise-D to begin her internship. Doctor Beverly Crusher acted as her mentor, and arranged for her to be able to work in all the major departments.
Shortly after her arrival on the Enterprise-D, a series of inexplicable events occurred. The first was the sudden appearance of a litter of puppies in Rogers' quarters (shortly after Rogers had spoken to Commander Riker about her three dogs, or "zoo" as her parents called it). With a simple thought, Rogers made the puppies disappear again.
Two other strange events took place. In a shuttlebay, Rogers had to intervene to prevent a container from falling on Commander Riker. When Geordi La Forge gave Rogers a tour of main engineering, the warp core temperature increased so much that it resulted in a warp core breach and while the breach was in progress, Rogers managed to stop it.
When Captain Picard discussed the event with his senior officers, Q appeared. He told them that he was responsible for the warp core breach to test if Rogers was fully Q; he told Picard that her parents were not real Humans but Q who had died in an accident. Q informs the senior staff that he has been sent to not only to evaluate but to train Amanda as well because his previous experiences with humans. Q further says that if Amanda is not trained in the use of her powers then she could accidently destroy herself or others.
Amanda was initially resistant to learning about her true nature and disliked Q due to his rude behavior towards her. Later, however, Amanda agreed to work with Q. As such, Q began training her and Amanda was briefly seduced by the ability to indulge in practically anything she wanted. This included creating a romantic fantasy with Commander Riker and herself. And, while Amanda could make him fall in love with her, she changed her mind. Amanda consistently demonstrated higher morals than Q himself as she questioned him about what the Q did with their powers, implying that she felt that if the Q had powers then they should be used to help others.
Picard, suspicious about the accident, told Data to investigate the deaths of Rogers' biological parents. It seemed that the circumstances were very rare; a tornado escaped the weather modification network and landed right on Rogers' house with a force which was inconsistent with the size of the tornado. Picard confronted Q with these facts, and Q acknowledged that Rogers' parents had been executed by the Continuum. If Rogers was not fully Q, she would be executed too, but if she was, he would take her back to the Continuum.
While talking to Picard, Q gave Rogers a choice: she could return with him to the Continuum, or she could refrain from using her Q powers. When Rogers chose the latter, the Enterprise-D received a distress call from Commander Riker, who was organizing the Enterprise-D's relief mission on Tagra IV. A reactor was overloading, threatening thousands of people. When La Forge told them that he could not fix the problem and meltdown was imminent, the problem seemed to resolve itself. Even the planet's atmospheric contamination and ecosystems were restored to their natural state.
Rogers could not let all those people die, so she used her Q powers. Acknowledging that she was Q, Amanda told Q that she would go with him but that first, she would need return to Earth to visit her adoptive parents and explain these events. She told Q, sternly, that he would have to be patient with her as she struggled to do this. Before leaving Rogers asked Dr. Crusher if she might come back one day to see her and then left the Enterprise-D with Q. (TNG: "True Q")
Background information Edit
Amanda, along with Junior, are the only children of Q known to exist.
In a 2011 interview, d'Abo commented, "Amanda had a vulnerability I related to because she was an orphan and in the episode was clearly at a stage in her life where she had some pretty important choices to make about her future. She had a strength and a wisdom about her. She was searching for the truth and searching for her own sense of truth. Though she was impressionable at times, it was inevitable to me through playing her that through her own self discovery she'd know which choices to make and what path to take." 
During the 0 Crisis, during which the USS Enterprise-E attempted to penetrate the galactic barrier, the female Q told Doctor Crusher that Amanda Rogers didn't "count" as a true child of the Q as she was conceived in a purely Human fashion, whereas "Junior" was conceived in the Q Continuum itself following "The Q and the Grey".
Amanda Rogers appears again in The Eternal Tide - referred to as "Q" by other Q - where she is shown to have formed a sibling-like relationship with "Junior". She is subsequently erased from history when trying to determine the source of an anomaly that is preventing her and Junior from traveling into the future by accessing an alternate timeline where USS Voyager was able to deal with the anomaly on its original trip home (before the events of "Endgame" changed history) only to be erased from existence as the resolution of the threat in that timeline also erased the Q from existence. Only Junior is shown to retain any memory of Amanda's existence.