My DS 7 has done ballet (and tap) since the age 4, he really enjoys it as do the girls in his class. I also dance at the school as an adult and have helped out in some of the children's classes.
The first few years are very much focused on learning the basic steps as well as learning some lovely little dance routines and a bit of character acting. My DS has done a couple of exams and also some rosettes which reward his progress.
Given how little she is, I'd suggest she tries both styles, they are so different. I can only comment on ballet, but know some girls drop out when they are very young as they have an idea in their head about what ballet is. They dream of being a prima ballerina on the stage, but the majority of the class is exercise based for specific steps.
I did consider gymnastics for my DS, but found our local club very hardcore looking for children for their elite programme and the trainers were very gruff. Facilities were amazing, but we got a feel straight away from the place and it wasn't for us.
In contrast, our ballet principal has been working for 40+ years and she is loving yet strict with the teenagers and small handful of adults that dance with her. She is extremely nurturing and encouraging with the younger children, as are the other teachers at the school. Very different to the principal that used to make us cry and hit our legs in 1980s! 
I'm in a different area to where I was brought up, so did specific research on the local ballet schools and the teachers qualifications/reputation. There are some schools that I wouldn't consider at all, so you need to be careful. Please feel free to DM me if you wanted any advice on what to look out for.