What branch of the military goes to war first?

Answer 1

The Air Force and Navy will pound the area frst and then the Marines and Army will go in after area is bombarded. Assuming you mean the US Army, it is generally the marines which are actually deployed first, they may not be the first to go into action though (e.g. see: Operation Desert Shield / Desert Storm)

Answer 2

That depends on the mission the U.S. want's to achieve. Traditionally it is the Marines, but on smaller scale action it is the Army with a quick deploying reaction force. The Navy can be used as a show of force at times to act as a blockade. So it really depends.

Addendum:

In most recent years, this has been a mission of JSOC (Joint Special Operations Command). JSOC coordinates operations between all services, using elements of the US Navy and Air Force to shell and bomb enemy positions before committing ground and close air support troops.

Most recently, for example, the mission into Pakistan to take Osama bin Laden was a JSOC mission involving elements of SEAL Team Six (US Navy and Marine Corps), the US Army, and Air Force.

Answer 3

Usually the airforce is the first to cross into enemy territory at the outbreak of hosilities.