I read that the Jewish day commences in the evening based on Genesis 1:5, "And the evening and the morning were the first day." Does this mean that the Jewish date changes at sunset each day instead of at midnight, as in the international calendar? I would really be interested in understanding this, so thanks for your help in advance!

That is correct. According to the jewish calander the date changes about 18 minutes after sunset which is the time the stars start appearing in the sky. This is the reason that we celebrate the holidays starting from the night since that is the true beggining of the next day. Rabbi Eli Blitz