Translation:Three dogs are barking in the store.
They can both translate to "crying", but in different senses. 泣く means sobbing and 鳴く means an animal making a sound. In other words, it's 鳴くfor humans crying (and animals with in tears in their eyes) and 泣くis for animals crying (as in, chirping, howling, whining, etc.).
How do you work out whether they are barking, howling, yowling, whining or growling? To "cry" surely seems more like howling and whining. Barking has aggressive connotations, so I feel as though it's a bit weird using the same word for chirping birds. Does this word work for the most common/famous sound an animal makes? Bleet for sheep, neigh for horse (rather than whinny), meow for cat (rather than purr)... Only then would it make sense for the dogs to be barking because it is the most common noise we associate with doggos (whereas the most common object we associate with dogs is hats, apparently).
に is used with directional verbs (行く and 来る being the big ones, though there's a few others), and で is used for non-directional verb. The easiest way to think about it now is how you go TO the park, but you play IN the park (you go TO the store, but the dogs bark AT/IN the store).