Using two buttons to increase and decrease number

Projects Programming
2 
const int buttonPin = 2;  // the number of the pushbutton pin
const int buttonPin2 = 3;

Do you count uh-huh, two, three? Or do you count 1, 2, 3? Number ALL the related variables.

int buttonState = 0;         // variable for reading the pushbutton status

Which of the two switches would that be?

  pinMode(buttonPin, INPUT);
  pinMode(buttonPin, INPUT);

You are not using the internal pullup resistors, so you must have external resistors. How ARE the switches wired? Why are you not using the internal ones?