How to send a message to my phone that my Arduino system is on when turning on my Arduino
You can send an SMS using the Arduino GSM library. Here is a bare bones sketch that sends one pre-set SMS when it starts up, based on this tutorial:
#include &lt;GSM.h&gt;

#define PINNUMBER &quot;1234&quot;
#define REMOTENUMBER &quot;07123456789&quot;
#define TXTMESSAGE &quot;System started!&quot;

GSM gsmAccess;
GSM_SMS sms;

void setup() {
  boolean notConnected = true;

  while (notConnected) {
    if (gsmAccess.begin(PINNUMBER) == GSM_READY) {
      notConnected = false;
    } else {
      delay(1000);
    }
  }

  sms.beginSMS(REMOTENUM);
  sms.print(TXTMESSAGE);
  sms.endSMS();
}

void loop() {
}

If you can't or don't want to use the GSM library, it should also be possible to send an SMS by using the SoftwareSerial library to control your GSM shield using AT codes. Here's a tutorial on this approach, which is lighter weight but might be more fiddly to get working, especially if you need a PIN for your SIM card.