So there won't be a definitive best for most of these and there are a ton of preference based choices for you to make so this may seem broad. I'll pitch my thoughts and maybe it could help.
Alliance Races Overview:
Human - Extra trinket for stun
Dwarf - Removes bleeds, poisons, has dmg reduction and has extra crit
Night Elf - Shadowmeld, faster move speed while dead (great for body res plays), faster while stealthed
Gnome - Cleanse roots/slows on a 1 min CD, more resource pool (mana, energy)
Draenei: HoT and increased primary stats (int, agi, str)
Worgen - 2 min cd 40% sprint. extra crit
Pandaren - Melee range (5 yd maybe) incapacitation similar to a gouge or paralysis
Horde Races Overview:
Orc - Stun reduction, enrage, higher pet dmg
Undead - Fear cleanse, little bit of dmg from touch of the forsaken (or w/e it's called)
Tauren - More health, quick cast AoE stun, extra crit
Troll - Haste increase (3min cd), slow reduction
Blood Elf - AoE silence that also restores a portion of your resource pool
Goblin - Rocket jump, extra haste
Pandaren - Melee range (5 yd maybe) incapacitation similar to a gouge or paralysis
Alliance:
Priest- Human, Night Elf
Warlock- Human, Worgen
Mage- Human, Night Elf, Gnome (extra mana benefits arcane), Worgen
Druid- Situational, Worgen for FCing and Night Elf for mid fights (either works though since shadowmeld can be useful on FCs too)
Rogue- Human, Night Elf, Gnome, Worgen. This one is super preference based in my opinion but my favorite is Night Elf
Monk- See rogue and exclude Worgen. Again this one is preference based
Shaman- Dwarf, I have not played Draenei so perhaps the heal and extra intellect/agility could be beneficial but stoneform dmg reduction is quite nice
Hunter- Surv is OP play what you want
Probably Human, Night Elf, Gnome, Worgen on top though. Another preference based
Warrior- Human, Gnome, Worgen
Paladin- Human, Dwarf. Again, not familiar with Draenei heal output
Horde:
Priest- Tauren, Goblin, Undead
Warlock- Orc, Goblin
Mage- Blood Elf, most others are good too
Druid- Tauren
Rogue- Blood Elf, Goblin if you can make jump work but Blood Elf is the definitive best here
Monk- Blood Elf, Tauren
Shaman- Tauren, Goblin
Hunter- Blood Elf, Tauren, if BM gets better than Orc could be useful but I'd stick to the first two
Warrior- Blood Elf, Tauren, Goblin (Noticing a pattern?)
Paladin- Tauren, Ret and Prot are not very good so you'll see Holy the most which Tauren will be better for