BlameTheParrot Very informative, and for those who are new to the Special Events the basics here apply. You will be given a Gift Icon to click on, then someone to speak to in your camp. Usually after completing the task of the first mine you will be asked to make a recipe in your kitchen, but be certain to make the one that doesn't have the Special Item (the Special Item is the one you use to buy some of the New items in the Shop) in it that you are gathering for the Achievement, otherwise not only will you have needlessly spent that Special Item but also still have to make the correct recipe for the task to register as being completed. During the Main Stroyline quests you will be asked to visit one or more of the Challenge areas, and as stated you will only need to pop your head in but not actually do them at that time. Fulfilling the Achievement always involves collecting a certain amount of the Special Item, and almost always involves having to get some of it in one or more of the Challenge areas.
As a sidenote, something you might want to do is to not unlock the recipe for the Special Event that has the Special Item as one of its ingredients. That way you will avoid accidentally using up a Special Item when you don't want to. Only unlock it when/if you actually want to make that recipe.