Question about times for first time flier.

Hi everyone! This is my first time flying ever in my life and I am prepared for everything else but there’s only one question that’s really been bugging me and I am not completely sure on.

I have a connecting flight through Istanbul both on the way to Italy and back. I wanted to know if the times presented on my flight are automatically updated to the time of where I am located. Say the departure time in Italy saying 2:05 pm so my departure time is at 2:05 pm based off Italy’s time rather than the place where I started flying in CST time. Is this correct? Last thing I want is to accidentally miss a connecting flight or return flight because I messed this up. I appreciate any advice given!