Unlike today's military, there was no set routing for recruits. A conscript from Augusta would probably have been sent to camp of instruction in Savannah, but a volunteer could have gone (or be sent) almost anywhere.
An independent company isn't regularly assigned to a regiment or battalion. This was true of most artillery companies. In 1861-62, most independent artillery companies were assigned to a brigade. Batteries in excess of the number required were assigned to reserve battalions like Nelson's. Eventually all batteries were grouped into artillery battalions and assigned to army divisions (an assembly of three to five brigades). Again, excess artillery battalions were often described as "reserve" battalions.
Here are two battlefield markers for Milledge's Battery. On the first, for the Battle of Sharpsburg MD (Antietam), click on the battery, then click on the battalion, then click on the tablet --
Here's another battery marker for the Battle of Gettysburg --
Notice that guns used by the battery are fewer in number, but the same type.
All of the batteries assigned to the army were "light" field artillery drawn by six- or four-horse teams, as opposed to "heavy" artillery seen in masonry forts along the coast and other heavily defended points. Heavy artillery was often stationary.