Practice meditation regularly, or any other type of relaxation method. Look for signs, physical and emotional. If you think your spirit animal might be an animal lurking around in your neighborhood, look around for a specific animal that you've spotted multiple times, as this might be giving a huge hint.
Also check for emotional urges, as this may mean you've found what you've been looking for. Look in paintings, drawings, and even tattoos. If you're an artsy type of person, your sign just might be in some form of art. If anime, or abstract art is your favorite style, pay special attention to that type. If you value your magic practices and routines, you might find your spirit animal in a dream.
It may take time for you to figure out what the series of dreams mean, but be patient, it will happen eventually. Make sure you are open-minded about your guide, because he/she chooses you!