Each healer has its advantages and disadvantages.
Cleric is the powerful single target healer. Its buffs are long lasting and enhance survival ability, and the heal over time is quick to set up and easy to maintain. Cleric Cures are undoubtedly the most potent among the three healing classes, and Purifying Wind is an AoE debuff remover, which is pretty much one of the most amazing things a Cleric can do for you in dungeons with AoE spamming bosses, like Veninfang Trial. Cleric has an AoE heal, but usually does not use that in the presence of Bards/Shamans since they have superior AoE healing (meaning faster and easier to maintain considering Cleric is there keeping the tank hale). A Cleric also has very nice offensive ability, since Holy Smite is a long range AoE that Cleric can use on mobs without even needing to deselect the tank. It's a 50% stun, and you have a chance to link stuns if you spam it (the stun even works on bosses).
Bard is an amazing and interesting class. It can kite and solo so many mobs at a time, and has a series of short-lived buffs that are very versatile. Poem of Wind is a 15 minute buff that gives it the speed it needs to outrun pretty much all the enemies you need to. Its Cures are weak compared to Cleric or Shaman unless you have specific gear for Bard, but Music of Life is the main reason people want Bards. There, you have a triple stacking potent heal over time for all allies in range. The upside to this is that it works itself and is easily kept up, the downside is that it is a buff and certain bosses make Bard MUCH less effective by wiping buffs constantly. Bards pretty much buff and flash emergency Cures in teams, and not too much else. Another notable buff with all-around use is Songs of Illusion, which grants a ton of Evasion and Defense to all allies in range of the song. Not sure why they decided to make such a broken buff, but it allows Thief to tank when used and makes normally good tanks super amazing (in upper levels, actually allowing tanks to cap out their defense easily). Burning Metal is also a point of interest since it's an instacast, triple stacking AoE debuff on enemies that does powerful DoT and a drastic decrease in defense. That sounds scary, and it really is since it allows Bards to easily manipulate 25+ enemies easily and solo them in dungeons while running around. Basically, around bosses that do not constantly wipe buffs, a good Bard is invaluable.
Shaman is a really fun class. With good gear, it's heals are pretty much as powerful as Cleric. Shaman gives offensive long-lasting buffs, and can cast a buff that reflects damage. The totems are controversial. Some people like Totem of Grace since it's MP efficient, but others dislike it because they are slow and break easily to boss AoE attacks. There are two ways to avoid it. Oftentimes, the totem is only just in range of the AoE since Shamans might cast them where they normally stand to heal the tank and rest of the team. Placing it a few steps back allows the totem to reach the Shaman and part of the team while staying out of range of many a boss's reach. Another way to use the totem is to slap it down right after an AoE as a relief heal. The other totem has a chance to stun, which is also appealing. Shaman's offensive capabilities are underwhelming, but Rage of Lightning can be helpful by slowing down an enemies rate of attack. Drops of Heaven is the main selling point of a Shaman, since this can be used as a quick AoE heal where Bard's Music of Life might get wiped. Spamming it is an effective healing routine since it also has a chance to trigger a heal over time effect on each individual affected by it (so each person has a chance to get it, not all at once). Shaman has a callback that allows a revive in battle. People are disappointed with it since its cooldown is fairly long at ten minutes and it is less powerful of a revive, but it is still useful for close encounters and perhaps salvaging a boss battle. Perhaps not the best tool it could be, but better than not having it at all. The last thing to mention about Shaman for now is its gimmicky Puma transformation. It grants minimal Eva and a nice movement speed boost for ducking out of AoE casts from bosses, or reviving in an 8-man dungeon and running back if you get hit by a freak critical. You can also roar in Puma for if you stand still and hit the space bar, and there is nothing wrong about that.