How can I make push buttons react to being pressed instantly?
You have a few options, including using interrupts. But I believe the most important thing is: "do not use delay()" and you might not even need to use interrupts...

Have a look on this nice post from Nick Gammon: http://www.gammon.com.au/blink with some tips about non-blocking coding.

Also a very simple "Blink LED" example: https://www.arduino.cc/en/Tutorial/BlinkWithoutDelay