Bird power consists of three separate aspects, card level, hat level, and slingshot level. Card level can be raised by collecting feathers of the right color, each bird card shows how many feathers you need to collect to level up the card. Hat level is shown in the hat menu; the more expensive the hat, the higher level it is. Slingshot can be leveled up by collecting full hat sets; the more you have full sets, the better your slingshot is.
The combined power of these three aspects is shown in the top left corner of the card and by tapping that card you can see how the power is distributed. In the hat menu, there is a small icon with an “i” on the left, that shows all you need to know!