Minecraft wiki entry for Mining, associated text: Pickaxe
It is good to have a diamond or netherite pickaxe, as although nether quartz ore and nether gold ore do not require any specific tier to mine, ancient debris does require at least diamond tier. A good enchantment to have, on top of the usual mending, unbreaking, and efficiency, is fortune, as opposed to silk touch, so that you can repair the pickaxe with experience gained from quartz and gold ore.
Armor
It is good to have some sort of armor, with at least one piece being gold, to keep piglins from attacking you. Otherwise, it is good to have at least iron tier armor. As usually in the Nether, fire protection is useful. While mining underground, ghasts won't be a threat, but if one is using beds or TNT to mine, blast protection can still be useful.
Weapon
Weapons are good to have in any situation, but especially so in the nether. As usually, a sword of best material available with good enchantments is recommended; however, fire aspect will be less useful in the Nether as many enemies are immune to fire damage, while looting can be especially valuable. If a trident is used, it should have loyalty, as it can otherwise be hard to retrieve it in Nether's terrain.
Food
Food is always important for survival, so keep quite a bit on you.
Blocks
it is good to have blocks on you to be able to cross over large gaps, block any lava pockets you encounter, and tower up to hard to reach places.
Fire resistance potion
Fire resistance potions are a very useful safety precaution. In the case one would fall in a lava pit or be set on fire, one can drink the potion, which will negate all fire or lava damage. While not necessary, this item is very good to have.