I've been maining WHM so far. (I'm at the Lvl 59 part of the MSQ).
I haven't had any issues with running any part of the main story so far. Que times are awesome and I really enjoy healing. The only parts that feels a little... off are those instanced battles you get here and there. In most of them you have a side character that will heal you. You have soooo much heal power that you don't really get to actually HEAL. You mostly just spam your damage moves (Stone and Aero, maybe Holy) and if your HP gets low use regen or heal. Never had any issues.
Minor Spoiler for Heavnsward
I don't think they wrote the story with a healer in mind... I mean we get a dramatic death scene. Complete with holding the guy in your arms as he dies. It just feels weird watching the guy die, but you are in your WHM gear and saying at the screen "why don't you just heal or Rez the SOB?"
Something useful that might help in dungeons and trials is to have a macro that will have your damaging spells target the target of the person you are targeting <tt> (target of target). This will let you have your tank selected so you are ready to heal them while you're putting out your dps. Quick Rez is also a good macro to set up. (qucik cast then Rez). You can find guides on how to do these on YouTube as well as other places.