For me, it begins with an image, a title, or a first line--or some combination of them. Driving to work in the morning I observe things around me, which I tend to write haiku about, trying to remember them and write them down when I reach the office. My standards for haiku are fairly strict, so not many of them make it to paper.
If it's a title or first line, I tend to make it in some meter. Depending on what the subject matter will be, I think about the form. If it is something that will involve a turn of thought, I'm thinking sonnet. I'm pre-disposed to sonnets anyway, and so that's my first choice. I have to keep reminding myself there's more to poetry than sonnets. If the subject matter is something repetitive, I might think villanelle. If I think this is a short subject, I might go with a cinquain.
Since images usually aren't enough to carry the poem (except in haiku and maybe cinquains), I have to think long and hard about metaphors. How can I convey what I want to say through a metaphor. This is difficult to me. Metaphors do not come to mind often, and sometimes an image, a title, or a first line will languish for a long, long time till I can come up with a metaphor. That's true about the two main poems I've written this year. Both had gestation periods of a couple of years while I thought through the metaphors.
I then think of hard hitting, action words I can use. If I think the poem will be a rhyming one, I look for the hard hitting word for a line ender and see if there are enough rhyming words for it. If not, I figure on burying it in a line and use common rhyming words.
At some point in time I begin thinking of what I want the reader to take away. Am I telling a story? Trying to encourage? Paint a pretty picture? Teach? All of these will enter into my thoughts when, after gestation is complete, I spit the thing out on paper, usually in a fairly short time. After that, I look at each word and make sure it complies with Coleridge's "best words in the best order".
Well, that went way beyond your question on how we get started with a poem. No charge for the extra thoughts.
NDG