For a 13 year old girl?
1. No Omnis (at least not the giant, fuck-huge ones, smaller books get called omnibuses because of the definition of the word)
2. I'd go with smaller trades that aren't kid targeted but also don't get into seriously dark shit, at 13 I can assume cursing and a bit of blood and violence isn't the end of the world as long as it doesn't get truly fucked up.
3. What I'd rec.
"Superman for all seasons" by Jeph Loeb and Tim Sale, its a nice pleasant and heartfelt story with creative and beautiful art that showcases why Clark is who he is over the course of 4 seasons. There are plenty of other one and done Superman stories, but I haven't read them. All-star Supes is not a good choice
"Batman: Year One" by Frank Miller and David Mazzucchelli, Dark and gritty but not over the line for a 13 year old, may have some bad language I can't totally remember... but its short and sweet, the art is great, and its the beginning of the Bat.
"The Flash" (n52) by Francis Manapul and Brian Buccellato, Hardcore Flash fans have probably perfectly good reason to not like this run, but its a perfectly serviceable series by writing standards, and the art.... is FUCKING AWESOME, the exact kind of thing that captures the mind of a young teen. Theres multiple paperback editions or you could go for the hardcover that collects iirc the first 24 issues.
Wonder Woman by Perez, others have already said it and theres really no argument against it, although Azzarello and Chiangs run from n52 looks fantastic and had a decent story that really only annoyed oldschool WW fans.
Thats all I'm going to suggest for lack of knowledge, also I'm not sure what to rec from Marvel other than like, Simonsons Thor run (which is also in small trade paperbacks), or as another person said Runaways.
Ultimate Spider-Man would be good too, the Bendis dialogue loathed by many is probably something a teen wouldn't mind because its all quippy and talky like tv and movies can be.