I got my daughters their first bras from the girls' section of Target- then we moved into the Ladies' underwear section when they needed a larger cup size. Other stores (Sears, for example) also carry an assortment of bra styles in AA cups (often just labeled with the band size, no cup size listed). This includes some traditional "training bra" styles, along with sports bras, and some with underwires that look like "real bras" even though they're sized for breast buds. Justice (which used to be Limited Too) has a very good selection of very well made bras, sized from AA through B cup (although my daughters preferred the adult bra section when they outgrew A cups).
If she's "moving out of the training bra stage" because she's outgrowing them (breasts are bigger) rather than because "she wants a real bra because she wants to feel grown up" then check out ANY lingerie department of any store that carries women's clothing. There are plenty of bras out there that aren't too expensive that aren't too sexy or lacy looking. I can't avoid the expensive bras for DD1 (who's now in a hard-to-find size with a small back and large cup) but for A and B cups, there are lots of inexpensive options.