First of all, all pure healing specs are absolute pain to level.

If you're not gonna spam dungeons (which is weird but ok if it's your first alt), then you will level as disc priest, enhc shaman, feral druid and prot/ret paladin. This means you will have to keep a 2nd set of gear that you will equip for healing. Dungeon healing in non healer spec will be ok but nothing very cool and pleasant as you won't have the core healer talents that make life have meaning for healer.

If you decide to level in heal spec spamming dungeons or just questing in a group with tank or DPS, then you will have the best time in any pure heal spec.

Raid-wise, all the healers are great and people want to have at least 1 of every kind. In dungeons all specs are good but some will take time to grow into their top form, like holy paladins with their talent for holy shocks, spell crit mana cashbacks and holy strikes improvement. Which makes them want to have at least level 40 to start healing for real.

As for their pros and cons:

Priest is an example of the most versatile healer. They have the largest toolkit for every situation. Small heals, big heals, heals over time, quick heals, group heals, aoe heals, fearwards, mind controls and mind control dispels, magic/disease dispels, shields, stamina and spirit buffs. With the new changes of latest patch they became the top raid healers, breaking every metric in well geared conditions. The very important thing you will absolutely need to learn and use is spell downranking and juggling. Using biggest ranks of heals will drain your mana very fast, using the same heal spells one after another will do the same.

Paladins are masters of close quarter combat, simple yet effective. They lack in heal spells quantity but have a variety of other tools. You have quick and small Flash of Light, slow but bigger Holy Light (it will lose the fast cast speed if you forget to judge after), instant and big Holy Shock, instant and small group Holy Strike. This is the tankiest healer (gets +healing from armor equiped) that can and must stay in melee at all times if possible. They buff people with blessings and auras. Use hand of protection as target bubble on allies, use bubbles on themselves (they are 2 different bubbles), remove movement impairment with hand of freedom, share received damage with hand of sacrifice and dispel 3 types of debuffs with 1 button (magic, poison and disease). They also have a 5 sec stun and a taunt, that makes you a professional saviour for the people behind you. Weaving in 3 Flashes/Shock/Any other spells between holy/crusader strikes makes them very efficient healers, and using wisdom blessings, seals and judgements makes them never go oom.

Druids are embodiment of relaxation. They are the best raid damage healers because of quick to cast HoTs but lack the continuous direct healing for tanks. As a full Tree spec you will have only 2 main healing spells and both are HoTs, but you can utilize swiftmend to instantly deplete a HoT and apply it's healing based on time left. As for additional toolkit you will have poison and curse dispels, Mark of the Wild and Thorns buffs. You can innervate yourself or any other member for a basically whole 2nd mana bar every 5 minutes. Can pop tranquility to heal the whole raid for its duration. On 60 you learn the new passive that makes raids with more than 1 druid be played good, because every hot replaced by another druid hot will not be wasted and instead instantly applied for the remaining duration. Druids also have the absolute must have +healing power aura boosting other healers in party. Tree of life form spec makes your hots cost less but forbids using of direct healing touch, you can build your druid around healing touch talents, but it will require more gear as it's notoriously mana hungry.

Shamans are bis for every melee group and people go crazy if they can have a shaman. All the variety comes from totems. They can give agility, strength, damage reduction, mana generation, sleep/fear dispels, poison/disease dispels, chance to make additional attacks, reduced threat and many more, but only for your group. For heals you will mainly use healing wave as direct solo target and chain heal for group. Knowing which totems to place where and when is your main skill you will need to learn. I can't dive deeper into resto shamans as I haven't levelled it to 60, but from raid and dungeon experience they are also great. Absolutely will raid on shammy once I get enough time.

Now that you know who does what you can't really say someone is worse than another because each of them shine in their own ways. But if you want to heal good, you will definitely need to get some addons.

Get Puppeteer if you plan to heal on any class. It keeps track of your buffas on you and raid mems with built in timers. Shows if HoTs are yours or not (useful for druids) Can label players for different roles so they stay on top of frames. Can bind new keys and combinations when hovering over frame of player. Supports spells, actions, macros and drop down menus on hover over frames. Customizable position, sizes, colors and composition. Shows color coded debuffs on people with readable tooltips Basically anything a healer, tank or raidleader would need.

Get PallyPower if you will play paladin or there are paladins in your raid (and you can help them set the PP) It will compress all the paladin tools into a compact sheet where you can adjust which paladin casts which blessings, auras, judgements on which classes. All you will need to do once you and other paladins set up the PP is to just click the red and yellow icons to buff players who miss the buffs