The future evolution of animals will driven almost exclusively by human activity. Whether we look at wild animals, domestic livestock, or pets it is man that will have the greatest effect on their evolution and survival in the future.
Wild animals will be forced to adapt to living in shrinking habitats or to sharing their habitat with humans and the animals they bring with them. This will, through natural selection, lead to some larger animal species shrinking. The smaller individuals within each kind will be able to sustain themselves in smaller territories and with less available food giving them an advantage over their larger kin. Large predators will be devastated and many may become extinct as they become a threat to livestock and therefore a target for humans who protect that livestock. Small animals will adapt to living in human environments. This is already happening in many cities as hawks and falcons adapt to nesting on buildings and hunting pigeons. Raccoons, opossums, and even coyotes are learning to live in suburban areas. Deer are learning to thrive there as well.
Domestic live stock will change due to direct human intervention. Higher demand for food will lead to the development of faster growing, larger animals. Expansion into less hospitable environments will lead to the development of livestock with an increased tolerance for cold, heat, or other extreme climates. The poultry industry has been working for years to develop faster growing breeds of chickens and turkeys. Broiler chickens that used to be processed at three months are now ready to process are six weeks. Turkeys are bred for larger breasts and have also had their grow time cut by half. Cattle are being bred for lower birth weight yet faster growth. New breeds are being developed that resist disease, give more milk, grow bigger faster, or some combination of these traits and others.
Pets will change too, but their change will be driven by consumers' desire for the unusual and unique pets. Smaller dog breeds are popular with city dwellers and new miniature breeds are sure to be on the way. Hypoallergenic and/or hairless pets are a growing trend too. Exotic pets are gaining popularity so we will likely see adaptations of wild and exotic creatures bred for smaller size and more docile temperament for the pet market.
All this is, of course, speculation. But it is an interesting subject to think about and the possibilities are almost limitless.