Yeah, it's weird and takes a little getting used to. You hit the spell, target the character with the cursor and hit the spell again to "launch" it. Somewhat literally -- you fire out a little sphere in an arc that hits the character and heals them. You can also mouse over two people and both will get healed when you fire the spell; I'm pretty sure they both get healed for less than healing a single person. There's some glyphs you can get later to give you a third person, extend the range, etc. I haven't played a Priest but I assume the basics are the same.
Mystics can also drop healing spheres (you've probably seen them, sort of large pinkish bubbles) that people in the party can pick up and get a heal effect. So when you're fighting a boss, you can drop spheres and concentrate on actively healing the front-line fighters while the ranged people can take care of themselves by grabbing spheres.
To interact with people (invite, inspect, etc) you hit ESC to go into the "cursor mode" and right click on them. You can also click on names in chat to whisper.
Wow. Regular ol' Joph fan club in here.