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In a book that switches between third-person omniscient and first-person limited perspectives across its chapters, Albertine is the first who gets the "first person" treatment. This kind of sets us up to think she's going to be a more important focal point than she actually is, but her role as the kind of eyes and ears of the younger generation of Kashpaws is pretty important within the narrative.
When we meet her, Albertine has just found out about June's death. It seems that her mother waited to tell her until after the funeral, supposedly because she was sure that Albertine wouldn't be able to get away from school long enough to go. You see, Albertine had run away from home a few years prior to that and ended up in nursing school.
Basically, that information fail was just part of some basic mother-daughter psychological warfare that, according to Albertine, was pretty much par for the course between them. Apparently, things had been bad for some time, (hence the running away):
I'd been the one who'd really blocked my mother's plans for being pure. I'd forced her to work for money, keeping books, instead of pursuing tasks that would bring divine glory on her head. I'd caused her to live in a trailer near Grandma so that there would be someone to care for me. Later on, I'd provided her with years of grinding grief. I had gone through a long phase of wickedness and run away. Yet now that I was on the straight and narrow, things were even worse between us. (1.2.15)
So, it seems that Zelda might have considered taking the veil at some point, and Albertine's, er, interference in that plan (by virtue of her existence) meant that she and her mother were always at odds.
Albertine ends up having an affair with Henry Lamartine Junior, Lulu's son, when the two meet in the Fargo when Albertine arrived there (after she ran away). It doesn't seem that anything much comes from it, since she basically scares the daylights out of Henry when she tries to touch his back the next day, not knowing that PTSD has left his extra-squirrely.
From Lipsha Morrissey, we learn toward the end of the novel that Albertine has ultimately set her sights on becoming a doctor.