The Wampanoag Indian tribe, which is most commonly credited with saving the pilgrims' lives, were known for their excellence in catching lobsters, as well as their ability to collect clams. Traditional Wampanoag food often focused on shellfish. So, it's safe to say that pilgrims likely ate what would now be a pretty expensive dinner.
Bread was, and still is, a staple in the Northeast. It should come as no surprise that this common food showed up at tables during Thanksgiving. During the 1600s, the most common kind of bread eaten in the Plymouth area was sourdough, also known as "cheate bread." So, if you're going for authenticity, go for sourdough.
Deer were extremely common around the Plymouth area, which means that both pilgrims and natives were likely to hunt them. The venison would most likely have been prepared with indigenous berries such as currants, making it a pretty decadent meal for all those involved.
Plymouth had a plethora of different species of birds available for human consumption. Swans, geese and ducks were some of the more common animals that found themselves on dinner tables at the time. Edward Winslow, who was present at the first Thanksgiving, actually described a massive hunt involving different fowl at the time.
Milk that was produced was almost never used as a drink. More often than not, milk was used for the production of butter. To make it even stranger, cows didn't arrive in the US on the Mayflower. Any milk or cheese that was made would have had to have been made with goat's milk.
Corn was definitely used by Indians for sweet corn pudding during the time of the first Thanksgiving. However, what was most likely present wasn't the sweet stuff, but the more bland corn known as flint corn. Since Thanksgiving happened during the late fall or winter, the corn was most likely ground up and preserved instead of eaten off the cob.
Believe it or not, one of the first things that the pilgrims did upon reaching the New World was establish a local brewery. Beer was much more common than wine or liquor at time, which means that the pilgrims who could afford to booze up likely drank beer during the first couple of Thanksgivings.