Animal imagery plays a big part in The Bacchae. For one, Euripides constantly summons the image of the fawn. Fawn skins were one of main pieces of attire for your average Bacchant. Early in the play we see Tiresias and Cadmus all decked out in their fawn skins as they go off to pay tribute to Dionysus. The fawn could be seen as symbolic of freedom and innocence. We hear the Chorus also express a longing to dance again like a "fawn at play in the green joy of a meadow," after it's escaped "the hallowing huntsman and his racing hounds" (171). This imagery also ties into the motif of hunting that's threaded throughout the play, and highlights the contrast of man and nature.
Another animal that pops up a lot in The Bacchae is the bull. One example is when the Herdsman and his buddies try to apprehend Agave and the other Maenads. This turns out to be a really bad idea, because the women go crazy and rip the men's cattle apart with their bare hands. The Herdsman laments that his "great lordly bulls" were "dragged to the ground like carcasses" (119). This kind of ritual dismemberment was often a part of Dionysian rituals, and is perhaps the reason that the bull became symbolic of the god. Dionysus even appears to the beguiled Pentheus as a bull before leading the King to be dismembered himself. Pentheus says, "Now I'd say your head was horned. […] for certainly you've changed – oh into a bull" (178). So, why would Dionysus choose to appear as an animal that he seemed to be a fan of dismembering? Your guess is as good as ours.