Also, since I am a vanilla pleb, I have to ask: how are priests in vanilla? Checking out the internet everyone agrees that you level by killing mobs with your wand; that's does not sound that bad if I don't oom every two mobs. Endgame set items look great and they seem to be the de facto healers in general (at least until paladins get rolling).
Having levelled and played multiple priests, I can say that they are great and I'm going with a priest in Classic too. Priests has one of the best, if not the best sustainability of all classes. Played smart, you never need to eat or drink. Put PWS on yourself, cast Holy Fire or Mind Blast to pull, put SWP on the enemy and then wand - and your wands do decent damage thanks to Wand Specialization. With Spirit Tap, your mana regens quickly. If it's a weak mob (caster mob or low-level) skip SWP. Packs can be split easily with Mind Control and Psychic Scream - thin a big pack out by casting MC on one and let the other mobs kill it, then dot them and once they get to melee range cast Psychic Scream. While they are running around, Mind Blast + Wand one by one. At 40, you respec for full shadow to get Shadowform that boosts your defence and shadow spell damage. You'll still want Wand Spec ASAP because otherwise, you'll spend the majority of your time drinking since Mind Flay has a terrible DPM ratio.
You can find dungeon groups easily and healing far less demanding than tanking. With heal-over-time, fast heals, big heals, group heals and a ress, you can cover every possible scenario. Dungeons are absolutely worth doing because the quests give a ton of XP and many of them have good gear that will help to level. You can heal everything up to BRD as Shadow spec.
In PvP, we go down pretty fast because of wearing cloth and even Improved PWS doesn't last that long, especially since smart players will target healers first. But hide behind others and pump out heals and your side will win. As Shadow, you'll melt faces but have serious mana issues, like Boomkins and Ele shammies.
For raids, priests are always in demand though both paladins and shamans are even more so (because of blessings and totems). Raid healing is an art in itself and I won't go into it here. You can raid as Shadow but it's difficult. Shadow Priest DPS isn't that good on its own, but it can augment the DPS of warlocks, so guilds that have at least 4 good warlocks often bring a single shadow priest along - after the debuff limit goes from 8 to 16. And if a raid doesn't have 4 quality locks, the buff from a shadow priest isn't worth it - better to bring another fury warrior, mage or rogue along. Naturally, that only goes for hardcore guilds, casual guilds let people raid whatever.