Should army be capitalized in this sentence?

My friend's oldest brother is going into the army next week.
No, 'army' doesn't have to be capitalised in that sentence. When referring to a specific army name, though, the word is normally capitalised: the British Army.

Always capitalize the names of the U.S. military services: Army, Marine Corps, Navy, Air Force, Space Force, Coast Guard, National Guard, Army Reserves, Marine Corps Reserves and Navy Reserves. Do not capitalize “reserves” or “reserve component.”