add a function to JButton

Question 1

button.addActionListener(new ActionListener() {
    @Override
    public void actionPerformed(ActionEvent e) {
        System.out.println("I was clicked!");
    }
});

Will print out I was clicked! every time button is clicked.

For the first button put the following code into the actionPerformed method.

JFrame frame = new Calculator();
frame.setDefaultCloseOperation(JFrame.EXIT_ON_CLOSE);
frame.setVisible(true);

And for the second one you could do something like.

MathQuiz.this.setVisible(false);
MathQuiz.this.dispose();

Question 2

In aditition to Arrem's answer:

To exit the only the current form use

this.dispose();

To exit the complete application use

system.exit(0);

To open a new frame use

NewForm nf = new NewForm();

Question 3

I have updated the code. It was having few errors. Check the following corrected one. Hope this is what is expected.

import java.awt.*;
import java.awt.event.*;
import javax.swing.ImageIcon.*;
import java.awt.Font.*;
import javax.swing.*; //import java swing to use the Jframe etc



class MathQuiz extends JFrame implements ActionListener
{

JCheckBox answer1 = new JCheckBox(); //checkbozx for answer 1
JCheckBox answer2 = new JCheckBox(); //checkbozx for answer 2
JCheckBox answer3 = new JCheckBox(); //checkbozx for answer 3
JTextArea textArea = new JTextArea();//text area for question
JButton exit = new JButton(); // button to exit quiz
JButton calculator = new JButton(); // button to open calculator
JLabel image = new JLabel(); //adds background image

// Some names which will be associated with numbers.
int answer;
int questionnumber;


/**
   * This bit of code starts the process of showing the window.
**/
public MathQuiz()
{
    // Drop down to the code labeled 'initComponents' below.

    try {
        initComponents();
    } catch (Exception e) {}

}


/**
 * This bit of code is the first bit that's run.
**/
static public void main(String[] args)
{
    MathQuiz frame = new MathQuiz(); //

    // The following code puts the Window on your screen.

    frame.setVisible(true); //allows the jframe to be visible
    frame.showQuestion(1);//shows the first question
    frame.setSize(378, 383);//sets the size of the frame
}





/**
 * This bit of code sets up how the Window looks.
**/
public void initComponents() throws Exception
{

    //edits the checkboxes for the answers

    answer1.setSize(new Dimension(80, 20)); //set size of answer 1 checkbox
    answer1.setVisible(true); //sets visible to answer 1 checkbox
    answer1.setLocation(new Point(20, 180)); //set location answer 1 checkbox
Font font1 = new Font("Verdana", Font.BOLD, 14);
answer1.setFont(font1);
answer1.setForeground(Color.RED);

    answer2.setSize(new Dimension(80, 20)); //set size of answer 2 checkbox
    answer2.setVisible(true);//sets visible to answer 2 checkbox
    answer2.setLocation(new Point(20, 210)); //set location answer 2 checkbox
Font font2 = new Font("Verdana", Font.BOLD, 14);
answer2.setFont(font2);
answer2.setForeground(Color.RED);

    answer3.setSize(new Dimension(80, 20)); //set size of answer 3 checkbox
    answer3.setVisible(true);//sets visible to answer 3 checkbox
    answer3.setLocation(new Point(20, 240)); //set location answer 3 checkbox
Font font3 = new Font("Verdana", Font.BOLD, 14);
answer3.setFont(font3);
answer3.setForeground(Color.RED);

    //code to edit the text area on the JFrame
    textArea.setSize(new Dimension(330, 60)); //set size of text ares
    textArea.setLineWrap(true); 
    textArea.setVisible(true); //sets visible to text area
    textArea.setLocation(new Point(20, 50)); //set location of text area
    textArea.setEditable(false); 
    Font font4 = new Font("Verdana", Font.BOLD, 14);
textArea.setFont(font4);
textArea.setForeground(Color.RED);

//code to edit the exit button on the JFrame
    exit.setSize(new Dimension(104, 40)); //set size of button
    exit.setVisible(true); //sets visible to button
    exit.setLocation(new Point(230, 170)); //set location of button
    exit.setText("Exit");

//code to edit the exit button on the JFrame
    calculator.setSize(new Dimension(104, 40)); //set size of button
    calculator.setVisible(true); //sets visible to button
    calculator.setLocation(new Point(230, 220)); //set location of button
calculator.setText("Calculator");

//code to edit the background image on the JFrame
image.setIcon(new ImageIcon("maths.jpg"));
image.setVisible(true);
image.setSize(new Dimension(370, 350)); //set size of button


    setSize(new Dimension(430, 292));
    setResizable(false);
    getContentPane().setLayout(null);
    setTitle("Multiple Choice Math Quiz!");
    setLocation(new Point(200, 200));
    getContentPane().add(answer1);
    getContentPane().add(answer2);
    getContentPane().add(answer3);
    getContentPane().add(textArea);
    getContentPane().add(exit);
    getContentPane().add(calculator);
    getContentPane().add(image);


    // This makes sure you can shut the window by clicking
    // on the close cross.

    addWindowListener(new WindowAdapter() {
        public void windowClosing(WindowEvent e) {
            thisWindowClosing(e);
        }
    });

    // This makes sure the checkboxes can let the
    // program know when they've been clicked on.

    answer1.addActionListener(this);
    answer2.addActionListener(this);
    answer3.addActionListener(this);

    //Added by Subodh 
    calculator.addActionListener(this);
    exit.addActionListener(this);           

}


/**
 * This code happens when someone clicks on the
 * close window cross. It shuts the window down.
**/
void thisWindowClosing(WindowEvent e)
{
    setVisible(false);
    dispose();
    System.exit(0);
}


/**
 * This code happens when someone clicks an answer checkbox.
 * It checks the answer and responds appropriately.
**/
public void actionPerformed(ActionEvent ae)
{

    if (((ae.getSource()).equals(answer1)) && (answer == 1)) {
        correct();
    } else if (((ae.getSource()).equals(answer2)) && (answer == 2)) {
        correct();
    } else if (((ae.getSource()).equals(answer3)) && (answer == 3)) {
        correct();
    } else {
        wrong();
    }

    //Added by Subodh
    if (ae.getSource().equals(calculator))
    {
        JFrame myframe = new JFrame();
        myframe.setSize(300,300);
        myframe.setVisible(true);
        myframe.setTitle("New Frame");
    }
    if(ae.getSource().equals(exit))
    {
        this.dispose();
    }


}


/**
 * Code done if right answer given.
**/
private void correct() {

    textArea.setText("Correct!\n\n");
    showQuestion(questionnumber);

}


/**
 * Code done if wrong answer given.
**/
private void wrong() {


    textArea.setText("Wrong Answer!\n\n");
    showQuestion(questionnumber);

}


/**
 * Code to show questions.
**/
private void showQuestion(int question) {

    // Show a quote, depending on a number.

    switch (question) {

        case 1:
            textArea.setText(textArea.getText() + "Question 1) 50 + 154?");
            answer1.setText("104");
            answer2.setText("175");
            answer3.setText("204");
            answer = 3;
            questionnumber = 2;
            answer1.setSelected(false);
            answer2.setSelected(false);
            answer3.setSelected(false);
            break;
        case 2:
            textArea.setText(textArea.getText() + "Question 2) 75 / 3?");
            answer1.setText("25");
            answer2.setText("34");
            answer3.setText("225");
            answer = 1;
            questionnumber = 3;
            answer1.setSelected(false);
            answer2.setSelected(false);
            answer3.setSelected(false);
            break;
        case 3:
            textArea.setText(textArea.getText() + "Question 3) 12 * 12?");
            answer1.setText("128");
            answer2.setText("24");
            answer3.setText("144");
            answer = 3;
            questionnumber = 4;
            answer1.setSelected(false);
            answer2.setSelected(false);
            answer3.setSelected(false);
            break;
        case 4:
            textArea.setText(textArea.getText() + "Question 4) 1256 / 4?");
            answer1.setText("400");
            answer2.setText("378");
            answer3.setText("314");
            answer = 3;
            questionnumber = 5;
            answer1.setSelected(false);
            answer2.setSelected(false);
            answer3.setSelected(false);
            break;
        case 5:
            textArea.setText(textArea.getText() + "Question 5) 135 * 7?");
            answer1.setText("854");
            answer2.setText("945");
            answer3.setText("1045");
            answer = 2;
            questionnumber = 6;
            answer1.setSelected(false);
            answer2.setSelected(false);
            answer3.setSelected(false);
            break;
        case 6:
            textArea.setText(textArea.getText() + "Question 6) 100 / 4?");
            answer1.setText("25");
            answer2.setText("20");
            answer3.setText("30");
            answer = 1;
            questionnumber = 7;
            answer1.setSelected(false);
            answer2.setSelected(false);
            answer3.setSelected(false);
            break;
        case 7:
            textArea.setText(textArea.getText() + "Question 7) 100 / 10?");
            answer1.setText("20");
            answer2.setText("0");
            answer3.setText("10");
            answer = 3;
            questionnumber = 1;
            answer1.setSelected(false);
            answer2.setSelected(false);
            answer3.setSelected(false);
            break;


    } // End of switch

} // End of 'quiz'

// End of code.
}
//and this is the JFrame that i want to open when i click on the JButton on the above JFrame



  class CalculatorFrame
  {

 public static void main(String[] args)
 {
 JFrame frame  = new MathQuiz();
 frame.setDefaultCloseOperation(JFrame.EXIT_ON_CLOSE);
 frame.setVisible(true);
 }
}