Hardware for a new project (Arduino Mini, Wireless, LEDs)

You can use either bluetooth, wifi, ethernet, RF, or IR.

Which ever is cheaper. This forum probably has hundreds of simple samples, i've got a few myself that I can give you.

But first try to write a code that can turn an LED on and off when it receives a "H" or "L" from the serial monitor.

try this.

LEDpin = 13;  //on-board LED pin 13

void setup() {
pinMode(LEDpin, OUTPUT);
Serial.begin(9600);
}

void loop() {
    if(Serial.available() ) { // if you have data to be read

     char Data = Serial.read();  //get data
      if (Data == 'H' || Data =='h') {  //check if incoming data meets requirments to turn on LED  
         digitalWrite(LEDpin, HIGH); //turn on LED
      }

      else if(Data == 'L' || Data =='l') { //check if incoming data meets requirments to turn off LED  
         digitalWrite(LEDpin, LOW);//turn off LED
      }

      else { } //nothing
    }
}