It increases as soon as you begin to travel, but the increase is very, very small when the speeds are far from the speed of light. The factor involved in the change of mass is v^2/c^2 (so the square of your velocity divided by the square of the speed of light). If v is small compared to c, this ratio will be extremely close to zero, hence we don’t see any effects like mass increases when we travel in cars.
In planes, the changes due to special relativity are still very tiny, and a human being wouldn’t notice them, but if you put an atomic clock on a plane, the time stretching effect due to the speed of the plane will mean that the atomic clock that flew is out of synch with a clock on the ground.
As soon as you begin to accelerate your energy begins to increase but the amount is incredibly small. Since mass and energy are interconnected it could also be said that your mass increases. It’s a little tricky but the key thing to remember is this mass/energy equivalence.