I played a priest in vanilla. Downed MC/AQ20/ZG and 2 bosses into BWL. I made a warrior first and didn't really understand the game and played that for a long time so I started priest late. That said, I never once farmed consumables or anything. I sat at about ~90 fire resist for Ragnaros with the Onyxia Hide Cloak. Never had a problem really. I would have a few mana potions if I got a drop from a dungeon, but mainly just had items with Spirit. I found having the DMF trinket that basically gives you Innervate was amazing coupled with Disc priest talents and I rarely had to ask for an innervate. I was on of our top healers.
Talent build I used was: https://classicdb.ch/?talent#bxghsVVoofxxMc
Alternative build 1 (spell reach, more tank armor): https://classicdb.ch/?talent#bxghsVVoobxxcf
Alternate build 2 (Better renew, more tank armor): https://classicdb.ch/?talent#bxghsVVoorxxcc
Most people would probably argue against Improved Renew, but I used it in dungeons a lot so I didn't have to respec every other day. I argue that you don't need Inspiration because all the Holy priests have it and I don't recall it stacking so they would cover it anyway. You could also skip Improved Fortitude or discuss with the other priests so they can spend their points wisely. I was every caster's friend with Power Infusion and would provide it to either our Priest Class leader or my friend who was a Warlock so he could get that extra damage boost and get himself up on the meters.
Edit: As for leveling, I would do as others have suggested, DoT, Shield, and Wand. When I got to the 40s range, I was pulling 4-5 mobs (those damn Gnolls) on accident and would just keep Renew and Shield up while wanding all the things. Other than that, I was constantly doing dungeons as that is my preferential way of leveling in the game.