I will attempt to address your questions in order.

How did Experience soldiers, leaders, and tactitions...get divided up

The simple answer to this is desertion and loyalty. Robert E Lee was asked just prior to him leaving for Virginia to be the head of the United States Army. He declined the appointment and took up the equivalent role in the Army of Northern Virginia. Other military officers did the same thing, resigning commissions and deserting the US Army in favor of positions within the Confederate Army. There was no Davis "I choose Lee" vs Lincoln "Crap, well I choose McClellan" division. The Confederacy was formed two months before the war began, more than enough time to establish an army while it was still safe to cross the Potomac River (for example.) I'm sure there were also cases of Confederate soldiers already holding positions in Federal installations in the South that simply took the installation with them to the side of the Confederacy.

Did the Army just split up?

Not so much at the general ranks. The United States through the end of WW1 had a tradition of maintaining a small peace time army. Since WW2, there has been no time, nor interest to stand down the US military like it had been done in the past. When the war broke out, the militaries on both sides opened enlistments and conscripted soldiers to bolster their ranks. Most of the Armies were civilians prior to the war. But I'm sure there were numbers of soldiers that deserted with their officers.

How did they decide who was in charge of what and where?

The same as any other point in history: By availability, skill and to some extent who they were.

Who Organized Units?

States. If you look at the vast majority of Civil War leaders and the battles themselves, you see things like 1st Alabama, 7th West Virginia, 79th New York. My 4th great grandfather served with the 3rd Mississippi Calvary near the end of the war. There were federal units like the 1st Infantry Division, but the majority of the Army at the time was formed at the state level.

How did they lay out the details of this war with it being one nation fighting itself

The Civil War, in the eyes of the Confederacy was a war of independence. They saw themselves as a new separate nation, with its own national capitol, its own constitution and flag. They conducted the war in that mindset. The war in the eyes of the Federal Government was that these states were under the control of an armed rebellion. West Virginia became its own state in 1863 because, in the area that became the state was formed from, it was decided that the government in Richmond had A) vacated its offices and B) could not be occupied by the duly elected replacement government because of the rebellion.

You say that the war was inside a single country, but the reality was that this was a war between two nations, one recognized, the other not. They fought within the boundaries that once constituted the Federal government, but the region that was the Confederacy had no federal presence and was not under Washington DC's control.

To elaborate upon organization, the American Civil War was one of the last hurrahs for the old Napoleonic organization of armies.

As Cfx99 wrote, the basic unit of both sides was the regiment. These would be raised by the states: either by the state government recruiting from its citizens, or by private citizens assembling their own units and offering their services to their states as the commanding officers (usually with the rank of lieutenant colonel or colonel) of said units. They were numbered in chronological order of when they were mustered into service: so the 8th Illinois was mustered prior to the 23rd, etc. Note that this only applied inside their respective states, the 20th regiment of one state could well have been raised prior to the 5th of another. Theoretically, regiments were 1,000 men, but battle, disease, and desertion wore their numbers down quickly, and in addition, many times regiments went into battle having left a good portion of their men behind, guarding camp or lines of supply or communication. By the end of the war, some regiments were down to under 100 men in effective strength, especially on the Confederate side; but for the sake of simplicity, in the war, regiments until the late stages of the war tended to number 400 men.

From there, these regiments were organized into brigades, usually consisting of 3-5 regiments (giving a brigade strength of 1200-2000), though brigades existed with as low as 2 regiments or as high as 6-8. The Confederates, valuing the sovereignty of their states, often organized brigades with regiments mostly or exclusively from the same state; for instance, Hood's famous Texas Brigade primarily contained Texas regiments (though an Arkansas regiment was later added), and the Stonewall Brigade was made up exclusively of Virginia regiments, with plenty of other examples out there. The Union on the other hand typically just assembled brigades out of whatever regiments were available; for instance, the famed 20th Maine at Little Round Top was brigade-mates with the 83rd Pennsylvania, 44th New York, and 16th Michigan, and they were facing down Law's Alabaman brigade. Brigades were typically (and on paper) commanded by brigadier generals, though out of sheer coincidence there were many brigades commanded by colonels, lieutenant colonels, or even majors.

From there, brigades were organized into divisions. Divisions, prior to the introduction of the corps by Napoleon, had been the "self sufficient units" of many battles, and indeed, in the late 1800s and early 1900s, the division would replace the regiment as the basic building block of armies. Union divisions tended to contain 2-4 brigades (2400-8000), while Confederate divisions 3-6 (3600-12000), and the Confederates further beefed up their divisions and increased their flexibility by adding subordinate artillery batteries (4 guns for Confederate batteries, 6 for Union batteries) at the division level. Divisions were (and still are) typically commanded by major generals, though brigadier generals often commanded them due to division commanders dying in battle being a semi-frequent occurrence.

Divisions, in turn, were organized into corps, which is where perhaps the biggest differences in organization between the two sides were prominent. Union corps tended to be smaller, more maneuverable and compact units, often having two or three small divisions with an artillery reserve; Confederate corps, on the other hand, were much larger, much more powerful units entirely, often with three or four beefy divisions (augmented with artillery), as well as an artillery reserve. For example, at Gettysburg, each of the seven Union corps were all under 14,000 men in strength, with many hovering around the 10,000 mark; by contrast, all three Confederate corps were over 20,000 men. Union corps were denoted by roman numeral according to their number in federal service (i.e. XI Corps, and it would be the only XI Corps in the entire country), while Confederate corps were informally organized army to army with normal numbers (2nd Corps, and there could well be another 2nd Corps in another army). Corps were led, on the Union side, by major generals, and on the Confederate side by major or lieutenant generals.

And this finally brings us to armies, where the Union would have large armies composed of many small corps (the Union army at Gettysburg had 93,540 men and 360 guns in seven corps according to a muster roll I'll link below), while Confederate armies were of medium size and composed of fewer, larger corps (three Confederate corps at Gettysburg totaling 69,915 men and 280 guns). Union armies were named after rivers (Army of the Potomac, Army of the Tennessee) while Confederate armies were named after geographical regions (Army of Northern Virginia, Army of Tennessee, Army of Mississippi). Union armies were also led by major generals, with Grant as lieutenant general being commander of all armies (no general was supposed to outrank Washington if I recall correctly, whose original title had been general), and Confederate armies were led by full generals.

Gettysburg campaign muster rolls so you can take a look at the size differences between Union and Confederate brigades, divisions, and corps located here.