Ummm, I've always thought there was an "offical" order to the routes, based on how their organised in the start menu screen;
Emi -> Hanako -> Lilly -> Rin -> Shizune.
And Lilly's seem to be the "Official Canon" route, since hers is always highlighted in the library and is the only one who gets an epilogue. (Though personally I thought Lilly was too good for Hisao) (And also, the titles of her acts; "Past, Present, Future", that felt very encompassing of a person's life, as if it's "The One", all the other ones seem somewhat less important compared to that.)
(and also, in Lilly's route there's that rooftop lunch scene where All the girls got together minus Shizune and Misha, that was as close as the game got in bringing everyone together for a warm "fun fuzzy" scene, with the purpose of emphasising / dramatising the resulting heart flutter from so many cute girls notwithstanding)
But... I think your first route should always be the girl you want / like most.
I had this idea that the routes were better played like this:
Emi -> Rin -> Hanako -> Lilly -> Shizune
(as reflected in my fic)
I reasoned Emi was the "least serious", on account of her childish behaviour,
And Rin was "second least serious", on account of her bizarre character.
And they were all paired together that way! It was perfect! So.. I thought Emi and Rin were kind of like the "silly" routes in Final Fantasy 7, with that crazy ninja girl and the black dude.
(There was also the matter of the time frame; Emi's was seemingly the shortest, Rin second-shortest, Hanako third (or overlaps with Rin), Lilly fourth, and Shizune very very long, all the way past Tanabata and beyond)
But then.. I realised Rin's route is better off right before Shizune's, since all the "deep" Rin stuff would make someone feel really.. somewhat down. (well I didn't feel down, just other players in general) And that makes all the cheery and outlandish things in Shizune's route more fun and vibrant!
And I think Hanako's route Has to be played before Lilly's, since Hanako's story is more-or-less a subset of Lilly's. It doesn't feel right to go through so much of Hanako's story elements, and you end up with Lilly. It would feel like taking the "special first time feeling" out of playing Hanako's route next.
I just posted my opinions on the internet.
And now I'll start a potential flame war; Hanako's route was the hottest. </lame_joke>