My 2.5-year-old seems to be relatively indifferent to our dog. I want him to learn to care for him and help him learn empathy and respect through having a dog. Any thoughts on tactics I can use to help the process?
Speaking as a parent with a dog, three cats, some fish and snails, and a 5-yr-old and a 34-month-old, time will be your best friend here. For our kids, we focus more on treating the animals gently (no chasing the cats, no pulling the dog's ears, remember the pet is a living creature, etc) and less on caring for the pets.
Your best bet is to model the behavior you want to see. As your son sees you petting the dog, and interacting with him, and training him, he will see that this loving, gentle behavior is the default for how to treat pets in your family.
As ours have grown, they are showing more empathy and interest in caring for the pets, but it's a slow process. The younger child wants to help feed the dog, but that's because he loves scooping the food. He rarely pets any of the animals, and we usually just focus on making sure he's gentle when he does approach them. The older child completely ignored the pets until about a year ago, and now has decided the cats rock and the dog is not a pest but someone to snuggle.