The way to get good at healing (or any other role for that matter) in group content is by doing group content. If you can get a group of friends from your FC or linkshells to run dungeons together with you, then all the better. But just letting the other players in DF know you're new at the role is usually enough to encourage some consideration, like taking on a few enemies at a time rather than racing past to grab everything at once.
When you're first starting out and still getting the hang of things, it's ok to focus exclusively on healing, so that way you'll only need to target the members of your party (F1-F4 if you're on a keyboard. Up/Down buttons if you're using a controller. In either case, those are based on the order characters appear in your party list, so just keep watching the party list to see who's in need of healing.) You can heal the first trio of low level dungeons with nothing but a single "Protect" at the beginning of the run and then "Cure" on whoever's HP drops too low throughout the rest of the run. So while healing is certainly a very important role, it's not an overly complex one, at least not to start off with.
As you gain experience and confidence, you'll get better at gauging when heals are going to be needed and when you've got enough leeway to mix in some damage skills of your own. At that point, you can add in swapping back and forth between Cleric Stance for damage spells or turning it off to cast healing spells. Then as you move into the intermediate dungeons, there will be enemies that inflict poison or other debuffs that you'll need to remove with the Esuna spell. And you'll need to judge when to heal a single player with Cure and when the whole group needs a healing spell like Medica, or choose between Cure and Cure II based on how much MP you can afford to spend on them, and so on. So the role will rise in complexity as you move further into the game, but such complexity increases are gradual enough to allow for some learning runs.