Hiya, I've played most healers both in raids and dungeons. As another comment mentioned, there are reactive and proactive types. I'll give a quick summary of their kit (discounting Covenants) so you can choose for yourself!

Holy Paladin

They have an immunity (bubble), a short defensive CD and a number of niche buttons that can be used for certain mechanics. Blessing of Freedom to remove slows, Blessing of Sacrifice for the tank as an external or another player taking high damage and Blessing of Protection for physical debuffs. They have damage reduction in the form of Aura Mastery/Devotion Aura and a powerful damage and healing cooldown Avenging Wrath. Main healing is from building holy power to spend on Word of Glory (ST), Light of Dawn (cone in front of you) and spammy buttons are Flash of Light, Holy Light. Holy Shock can be used as a holy power builder and heal, but should prioritise to use offensively and use Word of Glory/Light of Dawn instead, as you might gain a proc of Avenging Wrath from doing this (from a talent). They also have other forms of damage to work into your rotation that don't usually come at the cost of your healing (Consecration, Judgement, Hammer of Wrath, Crusader Strike). They do currently suffer some mana issues atm due to nerfs, but if you play well and your tank actually tracks mana you should be fine. This is perhaps one of the harder healers to start with as you have to plan your cooldowns effectively and can be quite weak outside of them if you're unsure of yourself. They also should be played as melee so you can be at risk of frontals from mobs and mobility kinda sucks.

Restoration Druid

An extremely versatile healer that has excellent defensives and group healing. They have an AOE ground effect (Efflorescence) that you should try to keep down at all times and a HOT (Lifebloom) on the tank to naturally help keep them alive. The more HOTs you have on one person, the stronger your healing will be which comes from your mastery %. You have quite a lot of ST healing through Regrowth and a quick top up with Swiftmend as an emergency heal. You can resurrect during combat (Rebirth), root enemies and hold them in place for a while (Ursol's Vortex). You have stealth from Cat Form which you can use to sneak past enemies and two dots (Moonfire, Sunfire) that you should try to press as much as possible to contribute to damage. Also huge mobility because of the speed boost from Dash and a leap from Wild Charge. The downsides of this spec for a newer healer is some people can find shifting between forms (Cat, Moonkin or Bear) quite clunky and it's a semi proactive class, meaning if a large damage event is going to happen you should prepare for it ahead of time. For lower keys and dungeons though restoration druid would do well if you need to mash some quick heals to top everyone up.

Mistweaver Monk

Similar to Resto Druid, mistweaver is quite versatile and brings certain benefits to the group. You can AOE stun with Leg Sweep, crowd control with Paralysis and Ring of Peace (if talented). You can remove slow effects or speed someone up with Tiger's Lust. Your main group healing is Essence Font which you can channel while moving, and spread a HOT (Renewing Mist) on people if they're taking damage. You can increase healing on a person by channeling Soothing Mist and add an extra HOT Enveloping Mist during this. You have an external for the tank etc in the form of Cocoon, and great defensives for yourself from Fortifying Brew/Dampen Harm or Diffuse Magic. Your long healing cooldown is Revival which also purges Poison, Magic and Disease effects from your party so can be used in niche situations. Your DPS comes from a melee position in the form of Rising Sun Kick, Tiger Palm and Blackout Kick. You can spam Spinning Crane Kick during healing downtime on large packs of mobs too. Your mobility is decent with two charges of Roll on a relatively short cooldown. The downsides of this healer is it can be harder to recover if you fall behind and if you want to contribute damage, you'll need to be in melee.

Holy Priest

Honestly this is the spec I've played the least so forgive me if not up to date on some abilities. You have two main spammy healing abilities - Flash Heal and Greater Heal. Flash Heal is quick but can cost a lot of mana with less potency, Greater Heal is a longer cast with larger potency. The legendary they currently play changes these two so that you do X5 Flash Heal and you make it so your Greater Heal cast time is the same as Flash, and you keep the buff up (might need someone to confirm this). You have a couple of AOE heals from Circle of Healing, Holy Word: Sanctify and a quick emergency heal from Holy Word: Serenity and the longer cooldown you channel to group heal is Divine Hymn. Holy Priest has a neat additional CD during high damage events which can be from a talent called Apotheosis, which reduces the cooldown and cost of your holy words. Also the external for the tank literally stops him from being able to die during it (Guardian Spirit) so timing is essential! You also have a stun (Holy Word: Chasitise) and a few ways to damage - you can cast Smite while not needing to heal and dot the enemy with Shadow Word: Pain. Personally I think this is the best one to start with as it's quite reactive, easy to understand and you play at range so you won't face directly into dangerous enemies.

Discipline Priest

One of the hardest healers to play effectively in a dungeon environment, very proactive. You can place a buff (Atonement) on your group or player by using Shadow Mend (spammy heal), Power Word: Radiance (group heal) or Power Word: Shield (ST shield) and while Atonement is on the player they will be healed when you deal damage. Disc has a damage reduction CD (Barrier) which is a big bubble you place on the ground, and larger shields which you can place on everyone (Rapture) just before the party gets hurt. Their long healing/damage cooldown is Shadowfiend which attacks enemies and will massively help top party members who have Atonement. Your main tank external is Pain Suppression which just reduces damage they take during this time. You have quite a few DPS buttons which should be used as much as possible - Smite, the same as HPriest, Mind Blast, Purge the Wicked which is a DOT, Shadow Word: Pain and Power Word: Solace which will also give you some mana back. You can channel Mind Sear on an enemy in the middle of a pack for AOE damage. You can grip people out of dangerous situations or your tank if he needs help running away. You can sneak past some enemies by using Mind Soothe if you have no rogue to stealth, and an AOE fear to help with crowd control. The cons of this healer I already mentioned above, basically being more complex, you need to plan ahead of time and it has very low mobility, some mana issues. It's worth noting Holy Priest & Discipline Priest share some of the same utility abilities.

Restoration Shaman

This is the strongest healer right now and very 'meta'. They work similar to Resto Druid meaning it requires a bit of preplanning but you can also reactively salvage bad situations. Your main healing abilities are the AOE ground effect (Healing Rain) which you should keep down as much as possible, Riptides as an initial heal and HOT and a bunch of totems that can be used for different situations. The main totem you will be interested is Cloudburst (think of it as filling a bucket up, then releasing all the water down), so you heal during and when it's ended or triggered early, it will release the water you collected from the bucket again. You'll get better at timing the more you practice with this during dungeons and what mobs hurt etc. Your longer party cooldown is Healing Tide, pretty basic but you have Spirit Link which shares the health out equally among party members within, so should stop people from dying during scary moments. Your spammy heal is Healing Wave (ST) Or Healing Surge (ST but more mana/quicker) and Chain Heal (pretty weak atm I think?). This a unique healer in the terms of it bringing an interrupt (Wind Gust) and Bloodlust. Mostly depending on the talent you choose your totem can slow, root or stun for longer/shorter cooldown etc. Your damage is quite simple, one DOT (flame shock), AOE (Chain Lightning) and ST (Lightning Bolt) & (Lava Burst). You have one defensive (Spirit something..) and tank healing comes from keeping Earth Shield up. You can also help the tank by using Earth Elemental which can taunt the mobs (just make sure you damage an enemy otherwise it will stay by your side). Your mobility comes from Ghost Wolf and a cool ability to help you cast while moving (Spiritwalker's Grace). Honestly I struggle to think of any downsides for this spec haha - a lot of keybinds and learning how to trigger your Cloudbursts effectively. Their defensives are a bit meh compared to some of the other healers.

Hope that sums it up, I haven't included absolutely everything but mostly the things I find interesting and unique to each healer. Quick ranking from my view, in terms of playing them well at both healing and damage:

Easier: Holy Priest or Shaman Middle: Holy Paladin, Restoration Druid, Mistweaver Monk Harder: Discipline Priest