In the intelligence community Nathan Hale is often called "America's first spy". He was captured by the British in 1776 and executed, famously stating "I only regret that I have but one life to lose for my country."

Hale's mission was to go behind the British lines and report on their movements.

This was the same for almost all human intelligence (HUMINT) gathering in those days. When battles were fought in formations the most valuable piece of information was at what point the formation would be weakest so that it can be penetrated/flanked/or used to flank. When the force was too large to fight, it allowed units to avoid engagement or retreat and re-engage under more favorable conditions.

More famous "spies" during the Civil War weren't actually conducting espionage as much as sabotage. The Great Locomotive Chase as an example, for which the first Medal of Honor was awarded. These men were executed as spies regardless.

Information of simple troop movements back in those days sounds useless with all the things the US intelligence community can do nowadays, but don't discredit its value.

General Beauregard of the Confederacy credits a spy from Virginia for the victory at First Manassas (First Bull Run). This spy was Bettie Duvall, and she rode from Washington to Fairfax to pass on those troop movements. Individual spies, same as scouts, could move much faster than entire units. However, scouts were uniformed and watched carefully. Spies were not uniformed and integrated themselves into local life. This was the advantage of a spy.

Mostly, the spies were just eyes. They went into an environment, kept an eye on things, and reported anything deemed valuable enough. None of this was centralized.

Allen Pinkerton made the first attempt at centralizing intelligence. He built a counterintelligence (CI) network in Washington to counter Confederate HUMINT. He built up his own HUMINT operation in Richmond to keep eyes and ears open. However, its effectiveness is questionable. Pinkerton's intel was often bad and cost a few battles by convincing the commanders not to attack a force they thought was stronger, but wasn't.

TL;DR + Conclusion - It was mostly a de-centralized network of eyes and ears in public. The targeted information was primarily troop movements, though other information like important infrastructure, routes, goals, and unit readiness were also valuable.