There are a lot of magic school series, but that kind of goes against the "discover" requirement, since it's already well known enough to have a school for it.
I highly recommend Mahoutsukai no Insatsujo/ A Witch's Printing Office, though it's not very focused on magic use.
The Fate series has women who use magic, though men use it too. Start with Fate Stay/Night.
Gonna show my age, but Chobits fits this bill very well, though the magic part is a very slow burn (and is less "learned" than... uncovered). It's also debated whether it is actually magic or just technology, but imo it's very clearly magic.
Sousou no Frieren has multiple arcs based around (mostly women) mages learning magic. It's also one of the best mangas or animes in decades, imo.
Mai HiME is a classic magical girls series that has great characters and is literally all about them "discovering" their individual powers. I actually recommend the anime over the manga, which shifts the POV from a male protag to Mai, his love interest, who is the real, titular MC.