The concept of spirit animals is not a universal one, but it does come from multiple cultures and paths; so to better answer your question, you'd need to provide the context you believe a spirit animal is from.
Personally I don't like the term spirit animal as it is often used as an English term for an aspect of Native American Traditions, which isn't an open culture. Thus to use spirit animals in that sense, one would be partaking in what is called Cultural Appropriation and that isn't a good thing.
If you're going for a more modern concept that doesn't have ties to a culture or path, than you would probably have a better time saying you work with animal spirits. One thing to keep in mind is that not everyone believes that all spirits we come in contact with are meant to guide or teach us. One principal of modern spirit work that seems to be commonly accepted is that to truly honor the spirit you have encountered, you have to show it respect and ask them things, rather than telling them to do something for you (unless you work along the lines of Ceremonialism and its associated paths).
Back to the subject; typically when a spirit wants to make its presence known to you, they'll start popping up in your life through various different manners (examples are through physically seeing the animals, seeing them in dreams or visions, the name of an animal being featured prominently, and a ton of other ways). The key to knowing when a spirit is trying to contact you is to pay attention to your what your intuition is saying. If you have a strong, unwavering, or "nagging" feeling that is associated with a being you feel is not you, chances are that it is a spirit.
Once you have established that a spirit is trying to contact you, you can begin by responding and researching what kind of spirit or animal it is.
Some folk can talk to spirits directly (though it is hardly the same as talking face to face with another human), but the brain still needs to interpret the information it receives, just like in real life; the point is that even though there is a physical explanation, there is also the possibility of a spiritual one as well. It is up to you to learn how to differentiate the two as some things have to be experienced in order for you to learn them.
The majority of people do not directly communicate with spirits, so they have to use systematic divination or pay attention to their surroundings as spirits can communicate via your surroundings; again, there will be a physical and scientific explanation for everything, but the key is learning when it is also spiritual and when it is not.
Once you've communicated with the spirit, you can establish what kind of relationship you have with it and find out what purpose you and the spirit might have if you are to work with each other. At any point in time, there is no rush for it all because spirit work takes time and is something that gradually builds upon itself as you go through your life's experiences.
There are different ways to work with spirits, and you can receive advice from many different sources, but ultimately you will have to find out what works best for yourself and your practice.