The main question was why the sorceress was after Natasha. As Ultemecia said, Natasha was a "doorway" into different worlds that she could use to travel between worlds. Edea knew this as Natasha's mother was also a "doorway." Cid also knew.

Natasha's mother left her with her father (Cid) at the orphanage because she knew she was dying. Natasha stayed at the orphanage for a few weeks before she was accidentally transported to Earth when Edea was being possessed by Ultemecia. A part of Ultemecia was drawn to Natasha and taken to Earth with her.

When the gang is at Trabia Garden and they figure out their childhood, Squall suddenly remembers a weird girl who used to try and steal his Griever ring. The rest remember that it was Headmaster Cid's daughter. Squall looks at Natasha and remembers "Aria." Aria runs to confront Headmaster Cid, who confirms that she was in the orphanage because her mother died.

Meanwhile, Natasha (Aria) has been going to classes, training will Zell, and working at the library. Her crush on Daniel fizzles out when she realizes that he's not interested (or has a girlfriend, hadn't decided yet) and she begins falling for (GASP) Zell as they've been spending so much time together. Honestly, it was becoming pretty obvious those two would hook up :P

One training session, Natasha is trying to reach her limit break with Zell encouraging her. After she knocks out a t-rexaur for the first time, Natasha ends up kissing Zell. He's surprised, she begins to apologize, but then he kisses her. Then, instead of telling him how she feels, she asks him if this happens between friends. Zell is confused and agrees. Natasha runs away because I live for that drama.

Anyway, when the big fight happens between Galbadia and Balamb Garden, Natasha is told by Headmaster Cid to stay out of it since Edea knows who she is and could come after her. Natasha defies her dad and follows the others anyway (idiot). However, she is shocked when Ultemecia possesses Rinoa and Edea comes back to her senses.

After Edea and Cid reunite, they ask Natasha to come to the orphanage with them. She decides to go and says goodbye to everyone. She stays at the orphanage for months with Cid and Edea, during which she's not really a part of the game as it happens.

I hadn't figured out yet how I would fit the storyline of the game into Natasha's story, but I did have an ending in mind.

Basically, the crew figure out that the only way Natasha can get back home is during the fight with Ultemecia. When they nearly destroy her, Natasha will approach her and take her to her world (because at this point, she has learned to somewhat control her powers). That night, Natasha realized she is in love with Zell, and tells him.

After the long fight with Ultemecia, Natasha approaches her as planned. She lets her do her magic on her before submitting the final blow. Time compression happens, and the last scene is of Natasha opening her eyes and thinking 'I'm home.'