As a first approximation, saying la donna mangia panini, while being a legitimate way to say “the woman eats sandwiches”, sounds slightly wooden, like that's the only thing she eats, a bit like saying that a herbivore feeds on such and such. La donna mangia i panini would mostly mean that she is eating some particular sandwiches we already know about. Finally, la donna mangia dei panini is a possible way to say that she eats (some, generic) sandwiches. Of course, the same holds for la donna mangia -/le/delle mele.
A general coverage of the use in Italian of definite articles (i panini), indefinite articles (un panino), partitive articles (dei panini) or no article at all (panini) would require a book, including situations that, while not at all random, are nevertheless very idiomatic. For instance you often use definite article even with a generic noun, rather than one you have recently mentioned (for instance, I love coffee = Mi piace il caffè).
Sorry to say so, but just with Duolingo you'll never get even close to catching these nuances, said as one who's giving Duolingo a chance but is exasperated by its utter lack of explanations.