Brand new WoW player and I want to level a healer. What is the best healer of dungeons and battlegrounds? (I want to be goblin because they have good racial abilities, so either priest or shaman)

if you want your main focus to be healing i would recommend the priest, but for a more balanced choice with more options i would choose shaman, it's all about how you play.

I don't really heal often, but I have a couple low level healers. I do however tank high level and I have noticed I don't die as much when a druid or priest is healing. Druid/priest also have area heals that affect multiple targets and are all around more versatile and able to save a group about to wipe.