In order to obtain all 16 colors of wool, you'll first need to obtain 16 dyes in order to dye the wool. Any color of wool can be dyed any other color, for example: you can dye White Wool to be Orange Wool, or Purple Wool to be Green Wool, etc.
Heres all the ways to obtain all the different types of Dye! (Compiled from the Minecraft Wiki)
-Crafted from Bone Meal.
-Crafted from Lily of the Valley (Flower)
Light Gray Dye
-Crafted from Black Dye + 2x White Dye
-Crafted from Gray Dye + White Dye
-Crafted from Azure Bluet, Oxeye Daisy, or White Tulip. (Flowers)
-Crafted from Black Dye + White Dye
-Crafted from Ink Sacs
-Crafted from Wither Rose (Flower)
-Crafted from Beetroot
-Crafted from Poppy, Red Tulip, or Rose Bush (Flowers)
-Crafted from Red Dye + Yellow Dye
-Crafted from Orange Tulip (Flower)
-Crafted from Dandelion, or Sunflower (Flowers)
-Obtained from Village Mason's chest
-Crafted from Green Dye + White Dye
-Obtained from Smelting Sea Pickles
-Obtained from Smelting Cactus
-Obtained from Desert Village House chests
Light Blue Dye
-Crafted from Blue Dye + White Dye
-Crafted from Blue Orchid
-Crafted from Blue Dye + Green Dye
-Crafted from Lapis Lazuli
-Crafted from Cornflower (Flower)
-Crafted from Blue Dye + Red Dye
-Crafted from Purple Dye + Pink Dye
-Crafted from 2x Red Dye + Blue Dye + White Dye
-Crafted from Pink Dye + Red Dye + Blue Dye
-Crafted from Allium, or Lilac (Flowers)
-Crafted from Red Dye + White Dye
-Crafted from Pink Tulip, or Peony (Flowers)
-Crafted from Cocoa Beans
-Wandering Traders can also have random Dyes for sale for 3 emeralds.
Okay, now that you know all the ways to obtain all 16 dyes, you just need to craft them with any piece of wool. Wool is obtained by killing, or shearing sheep. It can also spawn naturally in the world around Villages and Pillager Outposts.
edit: Wool can also be crafted with 4 string.
If I were you, id clear your inventory and hold one of each piece of wool in your inventory. Then, after you've obtained all 16 colors, your achievement should pop!