...I'm very cautious in thinking "What's important to the character" is the most important question. What's important to the reader is a better question. What the characters feel and think may be important to the reader, heh. But these details we give are for the reader, to help build the whole scene for the reader. This may seem like a fine line, and I suppose that writing from the perspective of "What is important to the character," "What does the character see, hear, feel," and so forth will often naturally lead to the inclusion of details that will also interest the reader, gain the reader's attention.
What's important to the character should be important to the reader (especially if we're talking about a POV or MC). If it's not, then the writer isn't doing their job.