Being cisgender, what I've found myself doing first is LISTENING to transgender friends and other transgender public figures who are willing to share their life journeys. I just accept that it's an experience I will never have, but it doesn't diminish their humanity and it shouldn't impede their access to quality-of-life.
The other thing I try to do is educate myself about the legal and systemic barriers faced by transgender people, so I can be an ally when it comes to implementing the exact policies that will break down those barriers. From my observations, it seems as though the lack of trans-specific health care for transgender people is a major impediment to being able to live one's own authentic life.
And I definitely see what you mean about the historical parallels between gay & lesbian people (and bisexual people) fighting for equality. Especially since homosexual and bisexual Americans are *STILL* grappling with the systemic and sociological obstacles related to heteronormativity as our society's default.
We're stronger as a united community rather than as fractured subdivisions of our community.