Every raid wants Priests, and they want a lot of them. The Priest is the strongest healer in Classic, with the deepest toolkit: big efficient heals, Power Word: Shield to prevent damage before it lands, Renew for rolling heal over time, and the dispels and Fear Ward that keep a raid alive through mechanics. If you want to be the person who keeps everyone standing, this is the class.
What makes the Classic Priest unique is that your race actually changes how the class plays. Priests get racial spells on top of the normal kit, and two of them are genuinely build defining. Undead get Devouring Plague and the legendary Will of the Forsaken, Dwarves and the others get their own flavor, and these racials shift how you level and how you perform.
And do not sleep on Shadow. A Shadow Priest with Spirit Tap barely stops to drink while leveling, since killing a mob restores your mana, and at 60 a Shadow Priest brings Shadow Weaving and Vampiric utility that makes them a real raid contributor. Healing is the safe, wanted role, but Shadow is the fun one.