ok, start with looking at the blink without delay example in the IDE.
You have fallen into the standard beginners trap in Arduino of using 'delay'. This is sometimes used in tutorials because it is simple but it is not useful for any significant code. Delay means to stop for a period of time, it is blocking code. The microcontroller does nothing and therefore can not switch any other LED or check buttons etc.
You need to use millis timing to control your LEDs. That is explained in the example above. With millis timing you need to grasp the concept of loop. Your code in the loop function will repeat over and over again very fast. The faster the better. Every loop the microcontroller can check the time and see if you need to turn on or off any led. It allows you to set any 'delayTime' you like but you cant use 'delay'.
But even if we cut you the slack you need, and take what you meant to say, you probably would have been better off not leaving a wide open path by asking for
// LED_1: On for 2000, off for 500
digitalWrite(LED_1,HIGH);
delay(2000);
digitalWrite(LED_1,LOW);
delay(500);
// LED_2: On for 500, off for 2000
digitalWrite(LED_2,HIGH);
delay(500);
digitalWrite(LED_2,LOW);
delay(2000);
You can combine those two patterns with:
digitalWrite(LED_1,HIGH);
digitalWrite(LED_2,HIGH);
delay(500);
digitalWrite(LED_2,LOW); // LED_2 was on for 500
delay(1500);
digitalWrite(LED_1,LOW); // LED_1 was on for 2000
delay(500);
// LED_1 has been off for 500, LED_2 has been off for 2000