Your character automatically starts with 50% worse shout cooldown, effectively making your beginning shout cooldown 50% higher than what is listed.
Player shout cooldown overtime can be decreased by a decent amount by learning shouts from word walls and also unlocking the shouts using dragon souls. You can expect ~10% cooldown reduction from learning all words, and another ~25% by unlocking words (Latter assuming you've used 30 dragon souls to unlock various words, and learned all of the "free" ones from quests).
Player shout cooldown reduction is capped at 135%. You can check your shout cool down in game by using console "player.getav shoutrecoverymult" at start of game it is 1.5 and at cap it is 0.15, you can bypass the cap via wearing Greybeard robes. Each piece of the robes reduces the cap by 0.05.
Shouts now properly affect wards. You'll need higher tier wards to protect yourself against strong attacks. No more flicking a novice ward on your heavy armor warrior versus that dragon! This also goes for harmful player shouts also. Some other effects now also properly effect wards, for example spider venom
Some shouts have multiple effects with different magnitudes. In these cases, the first listed value will be for the first effect, and the second listed value will be for the second effect.
"Mag Scaling" and "Dur Scaling" refer to whether the perks that affect shout magnitude/duration will improve the effect of these shouts.
"Only with trait" means only the magnitude increase from wearing dragonplate/dragonscale armor will apply.
Cooldown and Duration are seconds.
Nord - 12%
Lovers Stone - 10%
Talos' Scaling Blessing - 10%-25% (Acquire 1-25 dragon souls, through any means)
Speech Perk - 10%
Gift of Gab (Bard's College) - 5%
Talos Amulet - 4% (doubled if worshipper, quadrupled if Champion)
Greybeard Robes - 30% (all 3 pieces) - full set of Greybeard robes can surpass shout CD cap completely, each piece essentially lowers cap by 5%
Ancient Nord Armor - 4% (per piece) - full set grants bonus 4% shout cd reduction - total of 20%
Crown of Tongues - 9%
Corrupted Psiijic Robes - 5%
Ahzidal Full Set - 16% total
Dragon Aspect Shout (3 words) - 5%
Nord Neck - 6%, 9% for bloody variant (bosmer only)
Konahriik Mask: 15%
Shout Mastery
Words of Power Learned ~11% max
Words of Power Unlocked ~25% max
Important takeaway from this: if you want to have shout cooldowns at 0, you'll need to be wearing the greybeard robes with most of the non-equipment bonuses listed above.
If using the optional Derived Attributes Redone plugin, when you increase health on level up you may also improve your shout cooldown. This reduces it by 1.25% per level.