Almost everyone got Emi's path first, myself included. To get Emi's path, there are literally two choices that trigger it. If you do the right thing on both of them, it doesn't matter WHAT the hell you did earlier or what path you were on before, you automatically do Emi's. However, if you miss either of the two choices, you can never get her. It's just as easy to get her as it is to avoid her.
My opinion, the Order you should do...Emi>Shizune>Hanako>Rin>Lilly.
Lilly's is arguably the best arc with the overall best feeling all the way through, while I think Rin's is the best just as a general standpoint. Emi's is a great start, and trust me, you'll want to get Shizune's out of the way asap.