Explore the rooms of the Manor and win mini-games. Usually, items of the particular collection are hidden in the same place, but finding all the items won't be very easy. Sometimes you will get a quest to look for a collection item in an unusual place.
To find out where you can find a certain collection item, press on it in the collections window. The "Get" button in the next window will take you to a room or character with the item you need.