Core Keeper

I found the easiest egg to acquire is the curious egg, sold by the bearded merchant. You can get him to move to your base by building a valid house (must have walls, door and bed placed, no bigger than 8x8) then placing the slime oil inside (dropped by Glurch the Abominous Mass). Hatching an egg requires an egg incubator connected to an electricity generator. It's better to attempt this later in the game since the eggs cost money. There are 8 color variations of the embertail you will need to collect then place in the same chest.

The Wiki is very helpful for this achievement. It goes over where you can find eggs and all the color variations.

https://core-keeper.fandom.com/wiki/Pets

Each pet has 8 color variations. I found the easiest eggs to farm are the Owl eggs in the Azeos' Wilderness. I spent a lot of time there looking for the Cherry Trees. You will probably find a lot of eggs while you search for those. I used the Stormbringer weapon as I went through that zone because it has no durability and will break all the breakable objects that drop various resources.

Happy to answers questions if I didn't cover something :)