What rules address whenever a service member passes an officer of a different branch (Army soldier walks by an MC/Navy/CG officer)?

Regardless of branch, if they're an Officer, you salute.

I have to ask, what are they teaching you in boot camp these days? This should be common knowledge.

AR 600-25 for soldiers.
1–5. Hand salutes and salutes with arms
a. For instructions on executing the hand salute, see FM 3–21.5, paragraph 4–4.
b. All Army personnel in uniform are required to salute when they meet and recognize persons entitled to the salute. Salutes will be exchanged between officers (commissioned and warrant) and enlisted personnel, and with personnel of the Armed Forces of the United States (Army, Navy, Air Force, Marine Corps, Coast Guard), the commissioned corps of the National Oceanic and Atmospheric Administration, and the commissioned corps of the Public Health Service entitled to the salute.
c. The junior person shall salute first. Accompanying the rendering of the hand salute with an appropriate greeting such as, "Good Morning, Sir" or "Good Morning, Ma’am" is encouraged. Personnel will not salute indoors except when reporting to a superior officer.
d. The practice of saluting officers in official vehicles (recognized individually by rank or identifying vehicle plates and/or flags) is considered an appropriate courtesy and will be observed. Salutes are not required to be rendered by or to personnel who are driving or riding in privately owned vehicles, except by gate guards who will render salutes to recognized officers in all vehicles unless duties are of such a nature as to make the salute impractical. When military personnel are acting as drivers of a moving vehicle, they should not initiate a salute.
e. It is customary to salute officers of friendly foreign nations when recognized as such. The commanding general, U.S. European Command; the commanding general, U.S. Army Europe and Seventh U.S. Army; commanding general, U.S. Forces Korea and Eighth U.S. Army; commanding general, U.S. Army, Pacific; and commanding general, U.S. Army, Southern Command, are delegated the authority to establish policies for recognition courtesies prevailing locally for foreign officials. Should inactivation eliminate any of these commands, the authority will pass down to the next level of command. This authority will not be delegated further.
f. The President of the United States, as the commander in chief, will be saluted by Army personnel in uniform.
g. Civilian personnel, to include civilian guards, are not required to render the hand salute to military personnel or other civilian personnel.