Read Michael Harner's book The Way of the Shaman to get started. A shaman can help you with this as well. Just be sure they are experienced. Harner's book will help you decide.
It depends on what you mean by spirit animal. every body has nine animal spirit helpers that change during your life time so there is little use getting one of these tattooed. On the other hand everyone also has a Totem spirit animal that remains constant during your lifetime so I guess that is what you mean. the only way of efficiently discovering these is at an advanced stage of MARU Center work. (google it!). As far as tattoos goes the best method is to use special tattoos at certain places on the body. You can google the image of the famous Scythian man with tattooed animals on his body for some clues in this respect. these are only clues however. The true science of the tattoo as an mode of generating spiritual power is still a closely guarded secret known only to a few traditionalist Witches and Druids such as myself---
Spirit animal or soul animal?