Question

what was the process of U.S. mobilization after the country entered the war?

Answer 1

Mobilization is a function of several statutes, all around who and what is called up, and who has authority to do it. The President, on his own authority, can mobilize the Reserve and the Guard for actions up to 30 days. After this, it requires Congressional authority, which is either for 90 days, 180 days, or "the duration."
In addition, mobilization could involve the Draft. If the action is approved by Congress, or war is declared by Congress, the Selective Service is activated. Within 90 days, a lottery can be held and the first inductees processed thru MEPS (military enlistment processing).
No nation has ever mobilized without going to war, so a national mobilization is considered a hostile act and a declaration of war.