At the very beginning of the war most Federal uniforms were "frock coats." These were knee-length, somewhat styledcoats that were lined and way too hot for summer usage. Other than insignia of rank, according to military specifications, brass insignia was mostly on the leather goods associated with the uniform. A cartridge box with a leather strap was hanging from the left shoulder, across the chest, and resting on the right hip. This was kept in place by a leather belt. On the cartridge box as a shiny brass oval with the letters US on it (for the regular army, that is). On the front of the strap that crossed the chest was another shiny brass circle with an eagle on it. As a belt buckle was another shiny brass oval with the letters US on it. On the head gear would be a Jaegar horn (hunter's horn - a curved horn which symbolized that the wearer hunted down the enemies of his country) along with a brass letter and a brass number. These represented company and regiment.
It didn't take too long for the soldier to realize all this brass reflected the sunlight pretty well and created easier targets for the enemy to shoot at. Thyerefore, as much of this stuff was removed as they could get away with. Also, the stylish frock coat was soon replaced by a very common sack coat which came in four sizes; size 1, 2, 3, and 4.
The main things that were actually "on" the uniform of the Union Army were indications of rank, usually attached to the sleeve or on the shoulder boards in the case of general officers. According to the article pasted below, there were actually three versions of the standard uniform, the "service and campaign" uniform, the "parade" uniform, and the fatigue uniform which actually became the most common version by the end of the war thanks to its practicality and its smaller price tag.
There were some variations to the blue colored coat; the green given to sharpshooters along with their black buttons compared to the usual brass or other metal buttons. There were also some differences in the piping on the coats whether they were artillery or other different branches of the service.