If you want to be a better healer, do a lot of dungeons once you unlock them at 15. Sub-thirty dungeons tend to have very little sporadic damage, so you have plenty of time to develop good technique. There is a tendency for new healers to tunnel vision onto the party window and play whack-a-mole. This is not ideal, though for the most part it will satisfy your role. The best advice I can give you is to grow accustomed to actually watching the fight. There are a ton of indicators in combat that let you know where the damage is likely to fall. You can see which DPS is moving too slow to get out of the AoE, or when the mob is winding up for a big attack on the tank. This allows you to be a proactive healer, rather than reactive, and better prioritize where your healing is most needed. Make use of the HP bars floating over each character's head! Of course, the party window (and focus target, enemy list, primary target) are very useful references, and I have them all arranged surrounding my character for ease of view, but it doesn't provide all of the pertinent information.

Also, quality of life tip: If you are attempting to heal people outside of your party, such as in a FATE, the best place to click on them to target them is the little HP bar over their heads. It's very hit or miss if you try to select them by clicking their body I've found, but the HP bar is quite reliable.