What the achievement asks
Feed Bumbles the Mildenhall Growth Formula, in Stormsong Valley. Five points, Alliance only, and it pays a pet: Bumbles.
The wording hides the real shape of it. This is a four step unlock followed by a daily quest you have to do seven times, so it is a week of work rather than an afternoon.
Step one: the amulet
At the Mildenhall Meadery, kill Flowing Honey. They have a low chance to drop the Annealed Honey Amulet, which starts everything.
That spot is worth going to rather than wandering: three of them stand together and come back every three or four minutes, though you will have to clear the Honey Crazed Maulers in between.
Several people farmed for hours with nothing, then had the amulet drop within minutes of finishing the zone quest line. If nothing is dropping, finish the Stormsong Valley story first.
Step two: the letter
Take the amulet to Boralus, to the shop called Scribes & Nobles in the northern part of the city, and speak to Rosaline Mildenhall. She tells you her story and gives you Rosaline's Letter.
She has several spawn points around the city. One of them is inside a building you cannot enter, at 58.1, 66.3: if she is there, target her with a macro and use your interact key rather than clicking.
/tar Rosaline MildenhallStep three: the ingredient
Bring the letter back to the Meadery and give it to Ancel Mildenhall, in the laboratory basement. He starts Bumbles the Bee, which sends you to the other laboratory, in the cave up the hill.
Raimond's Secret Ingredient is on the table inside. Take it back to Ancel.
Do not do this step in a group. If someone else loots the bottle first, you get the credit but the bottle stops responding to you, and the only way out is to abandon the quest and farm the eight vials again.
Step four: seven days
Ancel then offers Let's Bee Friends, and this is the part that takes a week. It is a daily: gather Vials of Honey Slime from the mobs outside, use them to make the Mildenhall Growth Formula, and feed it to Bumbles, who is standing next to Ancel.
Once a day, seven days. Then the achievement and the pet.