According to AAA, the best time to travel by road is Thursday ( day of thanksgiving) and returning back on Sunday. For the lowest gas prices refer to gas buddy, it lets you see gas prices around the country based on the map.
A report by the Points Guy, Google Flights data determined the following as the trending destinations for Thanksgiving in 2017.
In the U.S.:
Amsterdam, the Netherlands
For the best airfare deals book your flights at least 21 days in advance.