Well, it varies... But usually I first select the paper to paint on - what type of paper and colour I think will suit the subject. Then I slap on some colours, just to get rid of that nasty big, blank space. Next I start to block in the basic shape of the animal, pausing often to walk back and view it from a distance.
Often I work on the eyes quite early, and get them fairly close to right. The eyes really are the "windows to the soul" and it's obvious when you get this wrong.

Sometimes I finish a painting in just a couple of sessions, but that's rare. More often it takes a few weeks of
2-3 sessions in the week, and putting it away for a few days, then bringing it out again for a fresh look at it. This usually points out some areas that need adjustment... so 4 weeks from go to whoa is fairly normal.