The most common cause of a squealing noise when starting a car is a failing drivetrain belt. When a belt becomes too worn, it loses its tension. The belt then becomes too loose to spin its assigned pulleys and instead rubs against them, creating the squeal.
In some cases, after the engine increases its idle speed, the worn belt grips and spins the pulleys, making the squeal dissipate. Belts can also become misaligned when worn, causing unnecessary friction on on the pulleys themselves. When belts become worn, the ridges that grip onto the grooves of the pulleys tend to flatten out, leading to a pulley misalignment. The pulley can also become misaligned for other reasons and change the direction of the belt, leading to the belt slipping. This will create more friction, leading to earlier belt failure.