What branch of the United States military is the oldest?

The ARMY was created before we became the United States. The Continental Army was founded on June 14, 1775, when the Continental Congress authorized enlistment of riflemen to serve the United Colonies for one year.

After the war against the British, the Army, Navy, and Marines where all eliminated, shelved due to lack of sufficient finances. It wasn't until 1790 that Congress brought back the Military branches with new names. In 1790 the First Branch formed was the Legion of the United States, the direct predecessor of the US Army and who would latter be called the U.S Army in 1798.

It has been suggested that the US Coast Guard was the first, but let's take a look at that claim briefly. In 1790 the Secretary of the Treasury established the maritime law enforcement service call the United States Revenue Service. The USRS remained in service until 1915 when it was merged with the US Life Saving Service and became the US Coast Guard. Technically, while the USCG can trace its roots to the USRS in 1790, just as the US Army can to the Continental Army and the Legion of the US, it did not actually exist until the 20th century. Additionally, the USRS remained under the authority of the Department of Treasury until its demise, so it was not a military branch. To be a military branch as opposed to a branch of law enforcement, it would have had to have been under the War Department or the Department of Defense.

The USCG like every other branch of the US military has a long history of traditions, and an equally long history of honor. Every branch should be respected as well as those who serve. However, the facts remain that the US Army was the first in 1798.
Technically the United States Coast Guard is not under any branch of the military, they fall under the Dept of Transportation. Only when a declaration of war is declared will the USCG fall under the DON (Dept of the Navy).