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Hey there! I'm glad I found this question...I am Persian, so hopefully you'll like my ideas :)
Because it's your first time, I'm going to suggest some dishes that first-timers usually enjoy, so it's nothing too crazy, but I hope you'll like it!
If you want to order an appetizer, I would suggest either the Shirazi salad (diced tomatoes, cucumbers, red onions, with parsley and lemon juice), or Mast-O-Khiar (plain yogurt with diced cucumbers and dried mint)
For the main dishes:
Chicken kabob (joojeh kabob), is a very popular dish. It consists of chicken breast meat, and is marinated in saffron (a popular persian spice) and lime juice. One of my favorites!
Filet mignon kabob (kabob barg) is also one of my very favoriters. It's basically like flattened filet mignon, and is very juicy and flavorful. Like the chicken, this is also marinated with saffron and lime juice.
Ground beef kabob (kabob koobideh) is another popular choice. Seasoned ground beef which is pressed around a skewer and barbecued, and is slid off the skewer and onto the plate. My boyfriend who is not Persian absolutely loves this dish.
+All dishes are served with long-grain rice and a grilled tomato (which you can mix in with the rice, if you want). I suggest you also sprinkle some sumac on the meat for extra flavor (the sumac is just a spice that is usually in a salt/pepper shaker...it has a dark reddish-purple color to it)
If you're feeling VERY adventurous, I suggest two stews:
Ghormeh sabzi - an herb stew with kideny beans and chunks of lamb.
Gheimeh - a lentil-tomato stew garnished with french fries.
Hope you like it!