You want an order? You start from left to right.
Emi -> Hanako -> Lilly -> Rin -> Shizune
Emi because the story is pretty much a simple, lighthearted romance story about growing up and moving on. Then you move to the story with the heavier issue dealing with facing your past to conquer the future, after that you reward yourself with a sweet--
but cliche--romance story between two people separated by two different worlds.
If you've spent your tissue at this point, then you might need to consider getting a new box or a new man-card--ask Kenji for advice.
The next two stories are deep, often requiring extra attention to little sub-stories in the background to fully understand what the hell is going on--I'll be honest, Rin and Shizune's story tire me out the fastest but I love their stories the most compare to Hanako, Lilly, or Emi's. You start out with Rin, put on your best intelligence card and your strongest man-card just in case you got the bad ending. Read carefully, analyze what's going on in Rin's whacked-out head, look at her paintings, etc. and see what she's trying to convey. You'll get your reward in the end with the beautiful ending. Shizune's story is full of humor, a good closure to all the feels roller coaster--UNLESS you wound up in the bad ending then you've just dug yourself a new hole to cry on. By far, Shizune's story requires the most attention, focus, and smarts in terms of reading it as large number of plot-points are scattered all across the story hinting you about who Shizune REALLY is, Misha's presence, and the relationship between them and Hisao. The ending is the perfect closure of their journey from the start of the year until the end and is further hinted that this type of relationship will last longer than others--sorry fellas', had to.
And by that, I declare that
Shizune's canon. 