That is an interesting question, since nearly every modern president has gone overseas to attend diplomatic meetings, lead a trade mission, or commemorate a particular event with other international leaders; years ago, these trips were conducted by ship, but in the modern era, Air Force One is the mode of transportation for the president. Some sources say both President Bush and President Clinton made the largest number of trips-- in two terms, they visited 74 different countries. In his first term, President Obama visited 35 countries. I enclose a useful link about presidential travel in the modern age.
It is difficult to determine which president used Air Force One the most, as usage can vary depending on individual circumstances and needs. Presidents who traveled extensively, such as Bill Clinton, George W. Bush, and Barack Obama, likely had higher usage during their terms. However, the exact number of flights and miles traveled by each president on Air Force One is not publicly available.
Obama
The Air Force One plane is primarily used by the President of the United States. It is the official aircraft for the President when traveling domestically or internationally.
bush
The president
Any aircraft the president boards instantly assumes the callsign "Air-force One"
President's political party
Air Force One
The President does not fly Air Force One- he flies ON Air Force One
Eleven Presidents have used different 'Air Force One's. This has been the Air Traffic designation of ANY aircraft the US President is flying in since 1953.
the president uses air force one because that air port is safe and secured for the president
The plane used to transport the President is always called Air Force One, irrespective of which particular plane he's in. It is generally a Boeing 747-200 modified with special equipment such in-flight refueling. To be more technical, it is Air Force One if he is on an Air Force Plane, Marine One when he is on the helicopter, or Navy One if he is on a Navy plane. The "one" gives the aircraft priority in aviation hierarchy (the smaller the number, the higher the priority).
Air Force One will remain Air Force One no matter who the President is.