Shadow Priests are very solid when leveling in Wrath of the Lich King and become much more stronger and efficient as you enter higher levels. Having talents such as Spirit Tap reduces your downtime drastically and having great defensive utility such as Fear and Power Word: Shield allows you to deal with any situations while leveling should they arise.
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Gameplay
Leveling as a Shadow Priest you can either kill 1 mob at a time which is generally slower but a very safe way of playing, or you can multi DoT and be fighting multiple enemies at the same time. By applying your 3 DoT spells Devouring Plague, Vampiric Touch and Shadow Word: Pain you can move onto another monster and do the same while using Power Word: Shield in-between to reduce damage taken from enemies.
Maintaining the Spirit Tap buff on you after killing a monster, alongside the Replenishment buff which gives you mana regeneration after casting Mind Blast (once you have Vampiric Touch talented) should always be prioritized. The Shadow Weaving buff should also be maintained as much as possible so you always have an extra damage boost. Shadowfiend should be used when your mana is running low as well as Dispersion to give you mana back and reduce the amount of time spent drinking. Keeping in mind Shadowfiend is your main damage cooldown as well as Dispersion being your main defensive ability.
Rotation
1-20
Mind Blast > Shadow Word Pain> Wand
Keeping your wand updated while leveling is crucial as it provides a huge amount of your damage until you reach higher levels.
20-40
Mind Blast > Devouring Plague > Shadow Word Pain > Wand
40-50
Mind Blast > Devouring Plague > Shadow Word Pain > Mind Flay
After reaching level 40 you can begin killing mobs with mostly your endgame rotation. You can continue to wand to finish mobs off if you won’t get the full value of your Mind Flay.
50-80
Vampiric Touch > Mind Blast > Devouring Plague > Shadow Word: Pain > Shadow Word: Death if enemy will die > Mind Blast > Mind Flay
Stat Priority
Pre level 40
Wand DPS > Spirit > Hit Rating > Spell Power > Intellect > Stamina > Crit/Haste
After level 40
Hit Rating > Spell Power > Spirit > Intellect > Stamina > Crit/Haste
Note that some stats aren’t obtainable until higher levels, but the above are the general stats to prioritize while leveling from 1 to 80
Glyphs
Major:
Glyph of Mind Flay, Glyph of Shadow
If you’re having mana issues you can replace one of the above glyphs with Glyph of Dispersion at level 60.