To transport your bird to the veterinarian, or anywhere else that is a short trip, it is usually best to use a specially made pet carrier, a smaller cage, or for small non-chewing birds, a heavy cardboard box with a sturdy lid, and air holes poked in it.
In winter, wrap the carrier in a blanket, and pre-warm the car before going out. To travel any distance (such as when moving), leave your bird in its regular cage, but lower the level of the water so it doesn’t slosh out, or give fruit pieces instead of water. Cover the cage with a blanket in winter, or a white sheet in summer (the cover helps prevent motion sickness caused by seeing out the windows).
Check the bird often, and try to get it to eat and drink whenever you stop. Carry water from home on any trip of more than two hours, as strange water might upset your bird’s system. Offer the bird full dishes of food and water immediately upon arrival, and give it lots of extra attention.
Never travel with your bird loose in the car! It could get out a window, or panic and get in the driver’s face, causing an accident. If you did have to swerve quickly or got in an accident, a loose bird would have no restraints or protection from flying objects, and probably would not survive.