This part of an ant's life is not well studied and documented, but you can extrapolate.
According to TheIncredibleAnt.com, ants travel at an average speed of 2 inches per second.
In one minute an ant will travel 10 feet or 120 inches. In one hour the ant will travel 600 feet or 7200 inches.
These insects don't seem to go tired but do exhibit torpor or short periods of inactivity. Torpor may be a strategy for conserving energy when there are no duties to perform.
According to Huber, in his The Natural History of Ants, ants become torpid only at 2° Reaum. below freezing (27° Fahrenheit), and apparently endeavor to preserve themselves from cold. When temperature is above this point, they follow their ordinary habits (he has seen them walk upon the snow).
From that statement, we can almost assume that ants walk or travel in its entire lifetime.
Taking into account the average life expectancy of an ant which is 45-60 days (this does not necessarily represent all ant species, others live a lot longer), an ant then with a life expectancy of 45 days will walk 648,000 feet (123 miles) and the one that lives for 60 days can travel about 864, 000 feet (164 miles).
Check out 10 Fascinating Facts About Ants from About.com.