My rough draft beginnings are purely a place to start writing, but by the second or third draft my stories are much more solid and I can think about where the real beginning should be - what will start the story off with the right tone and draw the reader in? I write suspenseful stories, so there needs to be something weird, creepy, or otherwise unsettling within the first couple of pages. My main character may not understand it, but his life is changing forever, and I want my reader to have a strong sense of that.
In thrillers and horror stories, this often involves getting a glimpse of the villain at the very beginning. This may be a paragraph or scene, or it may be done with point of view where we see our characters being watched by something icky. The second bit of this beginning is nearly always the daily life of the main characters - the mundane contrasting with the creepy to create a deep sense of foreboding on the part of the reader.
That second piece, the establishment of normal, is crucial. My reader needs to understand at least a little of what is status quo for my character so that the events to come have their proper impact. This establishment of normal may be only a paragraph, but it is essential, especially for stories of suspense.
As a reader, do you like to get a lot of action right up front, or do you prefer a bit of the normal before the rug is pulled out? Does the genre or tone/style of the story change your expectations for beginnings?