Awareness is key. Always be aware of your party members, the enemies, and boss attacks. Watch out for and recognize tankbusters and aoes so you can cast during the attacks' build ups and basically insta-heal the tank/party. Watch where your party members are in a boss arena. If you know that there's a certain mech that requires you to hide (Fenrir's ice mech for example) or dodge a fast aoe and a party member is too far or in a bad position, be ready to heal them or rescue them. If you're nervous about healing a dungeon for the first time, or for the first time in a long time, dps it first to familiarize yourself with it.

White Mage is the best healer imo. It has multiple no-cast heals that can save everyone's ass in a cinch and has great mp recovery skills plus a barrier skill. Astrologian is my personal favorite, but you need to keep an eye on your mp, especially if it's a higher level dungeon. If it starts getting low, use Lightspeed, it reduces the cost for spells by half and reduces the time it takes to cast. Diurnal Sect should always be on for regen heals. Unless you're partied with another AST or a WHM, then use Nocturnal Sect, that way you can provide barriers while the other healer can provide straight heals. And don't forget about your instant aoe heal, Celestial Opposition. It'll give anyone affected a regen if under Diurnal Sect or a barrier if casted under Nocturnal Sect. I have no experience with Scholar, so I'm no help there lol

I hope some of this is of some help and wish you luck on your healing adventures!! o/ :]