Does anyone know what part of the blueberry bush turns into the actual berry? Is it the flower, the bud, or something else? What should I be looking for?
The berry can develop when pollen is deposited on the stigma of the style in the pistil forming the Gynoecium or female part of the flower. The pollen then grows a tube that reaches the ovules within the ovary and the fertilized ovary develops into the fleshy fruit containing the seeds.
So if your blueberry bush flowers and the flowers get pollinated by bees (or more likely bumblebees) carrying pollen from flower to flower, the base of the flower can grow into blueberries.
Edit with photos as requested (click to enlarge).
Note the three future berries right in the centre: you can see they lost their petals, one style has dried and shrivelled, the other (younger) on the right is still light green. The ovum facing away fom the camera has started to swell to form the berry:
Here you see a cluster of immature berries, petals in various degrees of wilting and ovae transforming into berries.
If you buy blueberries you will sometimes find some remains of the style (like a thin thread in the center of the crown-shaped dimple) - see first picture - and dried petals that didn't fall off completely - see second picture.
And before you have to ask: it's the same for all types of fruit including nuts, from blueberries to pumpkins. But we do not only eat fruit, but also roots (think carrots, beets and radishes) and storage organs like bulbs (onions), tubers (potatoes, sunchokes), corms (taro) and rhizomes (ginger) or leaves (lettuce, spinach) of plants.