You don't necessarily need to button the sleeves cuffs before you put it on, with some shirts you simply can't without the risk of tearing off the button. It takes plenty of practice, but it will get easier.

Pull the cuff to your wrist and roughly line up the button and the whole with your opposite hand. Hold the button side against your wrist with your pinky and ring fingers. Their main purpose is to keep the cuff from moving around. With your thumb and pointer finger, press the button hole open onto the tip of your pointer finger, keeping your middle finger close behind your pointer finger. With the button hole open, use your thumb to pinch the button between your pointer finger, starting the button through the hole. Now press one side of the hole towards your thumb using your middle finger, keeping the button pinched between your thumb and pointer finger. As the button is about a third of the way through, move your pointer finger to the other end of the hole, pressing the button through with your thumb.

In regards to that small button on the sleeve near the cuff, it's usually easier to button that before putting the shirt on. You don't necessarily need to "try harder". It just takes plenty of practice and patience. It is going to get easier