While there are many parts to Coleridge's The Rime of the Ancient Mariner, the "most important" part is subjective. What this means is that each reader will come to find different parts of the poem important.
As for myself, I have always loved the part of the poem which offers the reader with the warning about behavior and consequences. While some people may not consider the life of an albatross as important, they must learn to look past the simplicity of a "simple bird's life." Instead, they need to consider that all life is interconnected in one way or another.
While some may argue that the poem does bring forth is a sense of greatness of nature and all living things, others may argue that the importance of the poem lies in the knowledge of "our" elders.