Answer: You either visit a shaman or skilled diviner and ask (the easy way out) or, you may pray to the Gods and ancestors to help you find your spirit animal or totem—and then pay attention to signs and omens and animals that cross your way: physically, in dreams, and in books and tv.
If you are called to more esoteric work yourself, you can engage in a trance journey or vision quest in the hopes of finding your animal but frankly, unless you’re a legitimate shaman, (and by legitimate shaman I do not mean a shamanic practitioner or harnerite), I wouldn’t go this route—not without guidance anyway. Sometimes a teacher might assign an apprentice or student to engage in the journey work and then, by all means, go for it. The teacher will have hopefully gauged that the student was ready for this type of work. Finding a spirit animal or totem means acknowledging an ally and that acknowledgement carries the responsibility for cultivating and properly maintaining the relationship. That can involve significant work for some people. One needs to understand that before trying to collect animal spirits like they were trading cards.