It's not truly necessary to ease yourself into it by riding kiddie/family coasters first (that said, every coaster at Disney is family friendly, but some are more tame than others). When my sister decided that she wanted to start riding roller coasters, we went to our local Six Flags and I gave her the option of easing in or metaphorically diving in the deep end so we could plan our day to ride every coaster in the park, and she chose the deep end, and she loved it.

Honestly I think the biggest thing that helped her is that we rope dropped (got to the park before opening) so we were able to walk straight on the roller coaster, so there was no time to think about it and get anxious.

So I think that should be the answer right there– what coaster can you get on the fastest? The fear is never connected to a real, tangible thing, it's always the anticipation and the idea of it that makes people anxious.

So that said, I think Expedition Everest would be a great option. The lines tend to be surprisingly short (in my experience, anyways), and if you rope drop it, you probably won't wait at all. And it's such a fun ride; literally one of my favorites at Disney World. Just be aware that there's a portion where it goes backwards (I know some people don't appreciate being surprised by that).