If you're looking for a print publication, try magazines and journals put out by related organizations, such as Animal Times (PETA), The UFAW Journal, The Animals' Voice, or Friends of Animals.
Many animal rights and welfare groups, like PETA, ASPCA, the International Society for Animal Rights, Animal Welfare Approved, and many more, also issue weekly or monthly email updates. If you'd rather read online, subscribe to related organizations' e-newsletters by signing up on their websites with your email address.