Minecraft wiki entry for Exploring_an_ancient_city, associated text: Multiple stacks of wool for blocking off vibrations.
A pair of shears to break wool quickly, and to collect the wool generated on the floor of corridors in case you run out of wool.
Some iron ingots to craft more shears.
A hoe with Silk Touch for collecting the various sculk blocks, and to destroy the very dangerous sculk shriekers.
The usual sword, pickaxe, and any other tools you usually take with you.
A significant supply of good food, like steak or golden carrots.
Some Night Vision potions.
Plenty of torches, at least two stacks.
A bed in case you die. (But do not place it too near or inside the Ancient City itself)
Some snowballs or eggs to distract the Warden in case it spawns. (The eggs can spawn in Chickens for extra distraction)
A bow, preferably enchanted with Infinity, to shoot arrows for distracting the warden. (Optional)
Some arrows, or just one if the bow has Infinity. (Optional)
A few milk buckets to remove the Darkness effect. (Optional)
Some iron blocks to construct some iron golems for fighting/distracting the warden. (Optional, not recommended)
Some snow blocks as a compact way to store snowballs, and to construct snow golems for distracting the warden. (Optional)
/ Some carved pumpkins to build snow golems/ iron golems for distracting the warden. (Optional)
Golden apples or enchanted golden apples to quickly heal. (Optional)
A full set of armor, ideally with Protection IV. (Optional)
It is not recommended to bring expensive tools or armor to the Deep Dark, since there are no regular hostile mobs that might attack you, and the warden is highly deadly in direct combat, even if the player has superior armor.