Picking the best race for the priest is difficult since this class gets unique spells based on its race, these affect how we rank the best drastically. For Horde for sure if you are going shadow you will want Undead, and Alliance can choose between Humans and Dwarves. With PVP in mind, we think Dwarves have an edge of humans, but they are both good choices so either will work.

1 Eye of Sulfuras And there you have it. Obviously Sulfuras is very much a guild-wide effort and the most common path to getting it is to have the guild gather up all the mats collectively, to then give to whoever earns the Eye - either through DKP or whichever method the guild uses. However, since pugs will be much more of a thing in Classic than they were in Vanilla, it's not completely insane to think someone will want to do this solo/without guild support, but for that you'd either have to have a whole lot of gold (not to mention luck to even get the Eye in a pug) and perhaps do some planning ahead of time, especially with the transmutes.
It is always worth ranking up any spells learned through talents or racials. Also, if there is some extra gold left in the ol’ coinpurse, consider buying Resurrection, Renew, Prayer of Healing, Fade, and Flash Heal. These spells are useful while healing dungeons, and worth the money if one chooses to do so. However, don’t break the bank for these spells. It is quite possible to get by without them, and a mount is worth more than a wider variety of healing spells.
Starshards: On-use channeled ability that deals Arcane damage. 30 second cooldown. If uninterrupted, this spell has the highest single target damage per mana of any priest spell in the game. Also deals slightly more damage per second than Mind Flay outside of Shadow Form. In terms of mana-efficiency, this spell can only be beaten by channeling Mind Flay on a target with 5 stacks of Shadow Weaving, but getting Shadow Weaving to 5 stacks is a waste of mana except against Elites and players.
Agility increases one’s armor, dodge chance, ranged attack power, and ability to crit with all weapons. I have been unable to confirm or deny whether this affects wands (everyone instantly calls you an idiot or a troll if you ask these kinds of questions). It is probably safe to assume that, as intended, priests do not want to prioritize Agility. However, it isn’t a totally useless stat, as armor and dodge chance are beneficial for leveling.
While actually hitting with a weapon isn’t important for a caster class like a shadow priest, the passive bonuses you’ll gain from maxing your weapon skill make them worth capping. Shadow priests benefit even less than other casters because two of their main abilities cannot be critical strikes. However, weapon skill does offer important defensive stats in the event you get hit by an add.
When that ass is firmly planted in the saddle of a sexy steed, feel free to spend as much money as desired. A lot of people suggest (and complain about) saving 1000 gold to buy an epic mount and the training necessary to use it. Don’t bother. The whole point of a mount is improving efficiency. Leveling from 40 to 60 with a mount is much faster than without, but in order to use an epic mount, one has to be level 60 anyway, and at that point, there is no need to be efficient. Raid bosses won’t die faster if you have your epic mount. They will die faster if the damage dealers aren’t dying because the priest could afford decent BoE gear at level 60.

In-Game Addon Version of My Leveling Guides:  As of June 6th the In-Game Addon version of my leveling guides has began construction!  It will be a long road before it's finished completely, so I cannot guarantee how much of it will be ready for Classic Launch but there is a decent chance that some starting zones will be complete for it by then.  It will be full of features, including:  waypoint arrows, automatic updating, and much more.  More features will be available for it overtime too.
Berserking: On-use ability that increases attack speed and cast speed by 10% to 25% depending on the priest’s health. At full health, the priest gains 10%, and can gain up to 25% if “badly hurt.” 3 minute cooldown. Very useful, as it increases the DPS of the priest’s wand, and allows the priest to cast spells faster. If used correctly, this can easily surprise a ganker when their health melts from the machinegun fire of wand blasts.
Quest helper addons like Questie are a must have and will make your leveling experience much smoother. Over the years, Blizzard has reworked WoW's questing interface to make it more user friendly and to present more information. In Classic, that was not the case. Quests were not readily displayed on the map, enemies did not have text to indicate you should kill them for a quest you had, and items did not have the glowing lines or sparkles around them to let you know that you should interact with them for a quest.