In recent years, farm animal welfare has become an important issue from both an animal science and a societal perspective. Moreover, livestock production is increasingly associated with negative consequences for human health, the environment, and the climate. Thus, there is clear evidence that a more sustainable focus in animal husbandry is needed. However, this is a complex and challenging task, as different stakeholders and emerging trade-offs regarding sustainability need to be considered. For example, outdoor climate housing systems may be socially valued and bring benefits to animal welfare, but at the same time may be detrimental to the environment, the climate, human health, or even animal health. With regard to a sustainable, future-oriented transformation of animal husbandry, it is crucial to identify potential trade-offs and to evaluate them from not only a scientific perspective, but also a societal perspective. Therefore, the aim of this study is to find out how citizens assess the importance of various aspects of livestock production from different goal categories (e.g., ensuring animal welfare vs. environmental or climate protection), as well as within the category “ensuring animal welfare” (e.g., good housing conditions vs. good health conditions). A total of 2,000 participants were recruited via an online panel provider using quota and split sampling. We used the best–worst scaling (BWS) approach to measure the importance of the different aspects for four different farm animal species (dairy cows: n = 503; fattening pigs: n = 500; broilers: n = 499; and laying hens: n = 498). The results show that, regardless of animal species, in a trade-off situation, citizens prioritize animal welfare and human health over other sustainability aspects and, in particular, over product attributes (i.e., product price and taste). In terms of animal welfare, the provision of good housing conditions was found to be the most important sub-aspect for all animal species, but good feeding and animal health were also ranked highly. We consider our results valuable for the development of policy frameworks aimed at a sustainable and socially accepted transformation of current animal husbandry.