There are many ways to solve this, you don't really link several applications together, you compile it all into one application and then build a flow for how you move through various states of the application. The best way to do this would be to approach it with OOP principles. For now to keep it easy, consider splitting up your draw functions and then using a variable to determine which one should be drawn.

In this program your first 2 screens work and you are all set up to put your pong game logic in. What I did was add an integer variable "currentScreen" that tells the program which screen the person is currently on.

1. int x = 125;
   int y = 235;
    
   PFont font;
   int currentScreen = 0;
    
   void setup(){
     size(400,400);
     smooth();
     font = loadFont("LucidaConsole-26.vlw");
     textFont(font);
   }
   
   void draw(){
     if(currentScreen == 0){
       drawStartScreen();
     } else if(currentScreen == 1){
       drawInstructionsScreen();
     } else {
       drawGameScreen();
     }
   }
    
   //your start screen
   void drawStartScreen(){
     background(0);
     textSize (26);
     fill (105,255,98);
     textAlign(CENTER);
    text("Welcome to Pong!",width/2,35);
    textSize(26);
    text("Start Game",width/2,150);
    text("Quit Game",width/2,225);
   }
   
   //your instructions screen
   void drawInstructionsScreen(){
     background(0);
     textSize (35);
     fill (105,255,98);
     textAlign(CENTER);
      text("Instructions",width/2,35);
      textSize(20);
      textAlign(LEFT);
      text("Use the up/down keys on the keyboard or mouse to control your paddle and stop the ball from going into your goal.\n\nFirst to score 15 is the winner.\n\nClick mouse to start game.",10,50,230,390);
     
     // Keyboard arrow keys icon
     //up/down arrow keys
     fill(105,255,98);
     noStroke();
     rect(315,50,35,35);
     rect(315,87,35,35);
     fill(0);
     triangle(332,52,348,66,317,66);
     rect(326,66,13,17);
     //down arrow
     triangle(332,120,317,106,348,106);
     rect(326,89,13,17);
     //Lefft right arrow keys
     stroke(105,255,98);
     rect(277,87,34,34);
     rect(353,87,34,34);
     fill(105,255,98);
     noStroke();
     triangle(279,104,296,89,296,119);//left arrow
     rect(296,94,13,20);
     triangle(385,104,370,119,370,89);//right arrow
     rect(357,94,13,20);
     //mouse direction
     //mouse up arrow
     noFill();
     stroke(105,255,98);
     triangle(332,150,317,164,348,164);
     rect(326,164,13,20);
     //mouse icon
     rect(317,189,32,25);
     line(333,189,333,214);
     arc(333,214,32,45,0,PI);
     //mouse down arrow
     triangle(332,276,317,262,348,262);
     rect(326,242,13,20);
   }
   
   void drawGameScreen(){
     //game logic here
   }
   
   
   void mousePressed(){
     if(currentScreen == 0){
       if ((mouseX > x) && (mouseX < 270) && (mouseY < y) && (mouseY > 210)){
        exit();
      } else {
        //make that an else-if to be specific that its on the button
        currentScreen = 1;
      }
     } else if(currentScreen == 1){
       currentScreen = 2;
     }
   }
   

hope that helps, best of luck!