Hi and congrats,
Regarding an authentic Mexican meal, there are 3 places that come to mind
La Granja, located close to the the airport (40 minutes away from the port) they have an awesome menu, that includes exotic dishes such as alligator, ants, ostrich, deer. The more traditional dishes are very good and the place looks like an old Hacienda. Expect to pay 25USd each way on a cab and about 50 USD for the meal
EL Cabrito, located next to the HArd Rock Cafe, (15 to 20 minutes away) this place has won International awards for their dishes, the decore is very typical of the region and well known for it''s goat dish, hand made tortillas, chips and all of the traditional dishes your heart will desire. Expect to pay 9 Usd for a cab and about 60 to 70 USD for the meal
El Zorrito or los Metates, located next to Galerias Diana Mall (10 to 15 minutes away) these 2 restaurants are side to side, a great menu of Mexican dishes including a meat lovers dish served in a lava rock dish known as molcajete, very good and enough for 2 persons. You will also find all of the traditional dishes at much more affordable prices, expect to pay about 30 to 40 USD. The only problem is that the waiters don't speak a very understandable English.
I strongly recommend that you ask the waiter for suggestions and try the "aguas frescas", refreshing drinks to accompany the meal, ( personally love Horchata, rice based drink) At el Cabrito, we enjoy a pitcher of watermelon.
Hope you enjoy your stay and congratulations again.