Cabinet members often state that they serve at the pleasure of the President, and the President can fire any Cabinet member at any time, but can the Cabinet members legally and Constitutionally ignore anything that the President tells them to do?
They ignore the president's will at their peril, because he can indeed dismiss them, apart from the vice president. For more information, see Wikipedia's article on the Tenure of Office Act.
It seems that the implication is, that the President nominates someone who will do what the President wants them to do, even if the President can't actually order them to do anything.
Why do you think the president can't order them to do anything? On the contrary, the president can order them to do lots of things. That's the essence of serving "at the pleasure of the president." If the president orders something they disagree with, they can resign, refuse to do it and see whether the president will fire them, or do it under protest.
Congress does delegate certain powers directly to various cabinet officers, but since the president retains the ability to dismiss those officers, the president exerts, or at least can exert, a significant degree of control over the conduct of cabinet officers.