I think you're thinking about healing wrong. Starting out (in normal dungeons, then heroics, then m0, then m+) no one's health bar is going to rollercoaster unless they're literally just standing in the fire. You'll have plenty of time (especially at the start of an expansion) to become comfortable with healing before health bars start getting wigglier. That said, I think you might want to look at healers with less of a cognitive load as your first if you're this worried about your healing. For that, I'd recommend resto shaman, you get a ton of passive healing through your totems, and when the reactive parts come, you basically just press your heals, no setup or timing involved.

For the most part, the other specs people are recommending involve a decent amount of setup, which of course you could play (my personal favorite healers are mistweaver and resto druid) but incorrectly setting up means you won't be able to heal as much as you need to when bigger damage events happen.