There's supposedly a learning curve involved when doing the dungeons. Sadly you can complete said dungeons perfectly fine without learning anything or even realising the curve is there. Dungeons from level 15 til 30 don't really teach you anything bar some extremely minor mechanics and may not even involve you as a healer. This dungeon is supposed to teach you (as a White Mage) to use Esuna. Although it somewhat fails in that department. Esuna is rather costly compared to Cure and unless the poison ticks for more damage than you can handle throughout the party, it's not worth removing the poison. If I remember correctly, it's perfectly fine to let someone build up 3 stacks before removing the poison.

The only real tip I can give you would be planning. Planning is everything when it comes to healing, as it involves mechanic handling and MP management. Take Cure and Cure II, for example. Cure II costs twice as much compared to Cure, but does not heal twice as much: 400 potency vs 650 potency. Cure also has a trait that may give you a free Cure II, whereas Cure II has no similar trait. Unless a lot of healing is required within a short time window, it's good practice to cast Cure over Cure II. This may change for endgame, however. It will depend on how aggressive you'd like to play later.

The following are included with planning:
Identifying mechanics
MP management
Movement mapping
party positioning
Spell tailoring*

Additionally, if you haven't done so yet, read all the tool tips concerning your spells, abilities and traits. This seems to be something that's occasionally being neglected.

* using a specific spell for specific situations/mechanics