I just got back into the game, making new characters, and want to try a healer. Reading about talents and spells, though, only gets me so far. How do these class specs actually play? How is playing a Resto shaman different from a disc priest, etc.? And which do you think is “most fun”?
Depending on when you’re returning from, either not much has changed, or quite a bit has changed. In general though, here’s about where the specs stand at the moment. Since I’m not sure what sort of content you’re interested in playing, I’ll just list them from a raid healer perspective.
As for me personally, I’m a Resto Druid main, so that’s my pick for “most fun”, but I’m biased because it’s my main. 
Druid : A Heal Over Time based class. Good at healing large groups, but is somewhat limited in raid utility cooldowns and big burst healing. Druid is a just a good “throughput” raid healer because it lacks utility cooldowns.
Monk : Monk has exceptionally good single target healing, but struggles more than some other healers with mana consumption. It is unique in that it can cast it’s biggest heals instantly while channeling a heal over time effect.
It’s limited cooldowns and strengths of being the highest mobility healer in the game are more suited to pvp healing than anything, but again they do have very good single target healing. They would feel most at home focusing on tank heals, and spot healing group members when the tank doesn’t need it.
Paladin: Paladins have some great utility cooldowns, and are still by and large the best tank healer, with their healing style these days being sort of a hybrid melee attacker/healer. Most of their healing comes from instant cast spells.
The closer they are to their target, the more healing they do, and especially with the new(old) holy power mechanic returning to them, they need to be up close and personal more than ever before in order to keep holy power generation rolling.
Priest - Discipline : This one could be a shock if you haven’t played since before Legion. The spec was reworked from the ground up at that point. The way it works now is the priest applies a buff called “atonement” to friendly targets, and then any damage the priest does, a portion of that damage is turned into healing for targets with atonement. It has the highest skill floor of all the healers in the game right now, with an even higher skill ceiling. That being said, when it finally clicks for you, it can pump out some truly impressive healing numbers, as well as having really good raid utility that priests are known for.
Priest - Holy : Holy is still the quintessential healer class. Healing is it’s main focus, and that’s what it does best. It has really good AoE healing for large groups, some pretty good raid utility, and works on a cooldown-reduction based play style. There are a couple really strong bigger heals that are on large cooldowns, but every one of your smaller heals reduces the cooldowns of the large heals. It’s not the most interesting healer to play, but it more than gets the job done.
And finally, Shaman : The totem master, especially moving into shadowlands, where it got a few more to use. Shaman has the most comprehensive raid utility and mitigation cooldown toolkit of any of the healers, and will always have an answer to a sticky situation. It has strong single target healing as well as good AoE healing, and is one of, if not the strongest raid healer in the game right now.
I haven’t played since before WotLK, so yeah, a lot has changed lol. In my day people always said Disc priest was only good for PvP.
I’m interested in at least trying all content. I liked dungeons and PvP as a kid, never really had a group dedicated enough for raiding, but I know people IRL that raid now and I might be able to get in with them, if I like it. From this and what I’ve read, it feels like Holy paladin is decent in both PvE and PvP, so I think that’s what I’ll try.
Quick reply:
Shaman - great for M+ and raids. Pretty good for PvP. They are strong and offer great utility with totems and strong heals both ST and AoE. Coupled with a good healing CD in spirit link.
RDruid - Hots hots hots. Lots of HoTs. Good for M+, average for raiding, meh for PvP. For group sustain damage intake theh can hot and watch the numbers roll. Also great utility with affinities and decent damage.
Priest - Disc - Learning curve. Practice and you can be amazing in raids and M+. Pretty good for small group fights in PvP and arena but feel lacking (from what I’ve seen) in bigger team fights.
Priest - Holy - The fantasy healing spec. Can feel light in M+ but get the job done. Great sustain healers in Raids to top off and provide burst AoE heals. Can get unlucky in PvP if focused and locked as they only have like 3 good insta heals.
Paladin - Strong for all content but the playstyle (melee heals) is difficult to master.
Monk - no.
Tried Disc priest as a class trial and I dig it. Feels like the Arc Mage from the Warhammer MMO (flawed game, but had some very unique classes).