EDIT: Ha, this got long; I'll probably be adding another post or two to this in the next couple days, but feel free to as any follow up questions as they come.

Honestly, not a whole lot changed on the fundamental, tactical level; it was their ability to execute the tactics that changed. The basic forces at play -infantry with muzzleloading blackpowder muskets and bayonets, direct firing field artillery, cavalry armed with carbines, pistols and sabres- were the same in 1865 as in 1709. It still took twenty or thirty seconds to reload after every shot, men still only carried sixty rounds, troops still ran when faced with a bayonet charge, most soldiers had little hope of hitting anything past a hundred yards, muskets and artillery still threw out thick clouds of smoke with every shot, leaving soldiers deaf and blind, and everyone knew from the beginning that it was good to have field fortifications and bad for your enemy to have them. Joe Johnston dug a trench line in Northern Virginia after First Bull Run, McClellan dug his way all the way up the York James Peninsula, and even after four years of bloodshed, commanders on both sides still formed their men up in dense columns to frontally assault these fortifications, because sometimes that's just what you have to do.

The most important thing to remember is that American military officers had no frame of reference for a conflict on this scale. The U.S. Army consisted of about 16,000 officers and men at the beginning of the conflict, and their goal eventually became the outright conquest of a nation the size of continental Europe; the largest force ever commanded by an American officer up to that point had been the 12,500 men who Winfield Scott led in the Mexican War. Over the course of the war, the United States mobilized 2.5 million men for war, putting 10% of the population in arms; the Confederacy armed nearly a million men to fight for independence, a staggering 20% of the population.

Naturally, neither side had enough professional officers to control these armies with much skill. The professional officers they did have were graduates of West Point, which was fundamentally an engineering college; the best scoring cadets commissioned into the Corps of Engineers, followed by Artillery and Infantry, with Cavalry in dead last. Controlling large bodies of troops -divisions and corps, field armies, tens of thousands of men- was not part of their professional universe. One of Lee's corps commanders, Richard S. Ewell, mentioned that in the Army, "he had learned all about commanding fifty U.S. Dragoons, and had forgotten everything else."

Naturally, this was a problem. Commanding fifty troopers is one thing, but armies of tens of thousands require, for instance, a million pounds of food and forage a day. Up to a million rifle cartridges for every day of combat. Thousands of artillery rounds, replacement parts for rifles, cannons, wagons. Men wear out their shoes marching 20 miles a day; those have to be replaced. They need good maps; you can't send 90,000 men into battle down the same road, or the men at the back of the marching column won't even know there's a battle at the front. You have to split the army into multiple columns, have them take multiple paths in the same direction, but they have to all converge at a common point in case one of them runs into the enemy. And then there's the whole alchemy of military intelligence, so you know where you might run into the enemy.

Command at the highest levels is a highly technical exercise; not just anyone can do it, but even the high command had to cast a wide net. Philip Sheridan was a second lieutenant in 1861 and an army commander in 1864; ask any modern NCO if they'd trust a second lieutenant with a pair of safety scissors, much less the lives of tens of thousands. Sheridan was fortunately dealt a winning hand in the latter battles, but not everyone was so fortunate. Men who held the rank of colonel in 1862 did not know the command for their 800 men to deploy from column into line; training tens of thousands of men to brave the hurricane of musketry and artillery to press an attack with the bayonet was just not something they were able to do. This could have devastating consequences. In The Battle of Gettysburg, during Pickett's Charge, many of the regiments involved in the assault stopped to trade shots with an enemy behind a stone wall, when if they'd just swept in with the bayonet, they'd have driven them from their positions; at the High Water Mark, one regiment did just that, and the Union had to rush in reserves to block them after the first line of defenders fled.

Modern armies have large staffs to manage the vast resources they consume in action; a Corps headquarters in the modern U.S. Army has about 300 officers and men in various sections and command posts. In the Napoleonic Wars, each of Napoleon's marshals had a staff of thirty men to assist them in running their corps. Robert E. Lee, commanding 75,000 men in the Army of Northern Virginia, had a grand total of one adjutant, one military secretary, five aides, and five clerks at his headquarters. When McClellan found a complete copy of Lee's campaign plan in 1862, before he could make any use of it, he had to figure out if it was part of a deception effort by the Confederates, weigh it against other intelligence reports, make sure it was still current, and identify and counter risks. He did this without any assistance from his staff, and could only develop plans, guide his subordinates, and issue orders after they were completed.

The professional education the army commanders had received at West Point had taught them that staff work did not require superfluous specialized training, and that any officer could undertake the associated tasks himself. This shortage of trained staff officers was a major disadvantage during operations. In addition to the McClellan example above, Lee cannibalized his own staff (twelve men, remember) to perform reconnaissance missions at the Battle of Gettysburg, which left him without aides at various points in the battle.

With desperate shortages of trained (to say nothing of skilled) commanders, and the near total absence of trained staffs, the most useful training the men and lower level officers got was experience on the battlefield, the most brutal of all teachers. It is also a capricious teacher; the men learn until they are killed, and because the Union generally didn't slot new recruits into existing units, instead raising new ones from scratch, these men had to learn from the beginning. Without replacements, veteran units that had fought in dozens of battles were just ground down into nothingness.

The quality of the Union soldier is believed to have peaked in 1863; most of the men there were veterans, and importantly, volunteers. The data shows that in the Civil War, the most battle casualties and fewest desertions came from volunteer units. This came after perhaps a quarter million casualties in 1862. After Gettysburg, and especially through 1864, an increasing number of Union troops were conscripts. During U.S. Grant's Overland campaign in 1864, losses were so high, he transferred artillerymen out of the forts around Washington, gave them crash infantry training, and put them on the firing line; their skill and morale left much to be desired. Beyond that, many of the regiments fighting in that campaign had been raised for three year terms in 1861, and would be leaving soon; this led to an unwillingness to take risks, undermining combat performance.

Compounding the low quality of the infantry and officer corps, the cavalry was essentially non existent in the classic sense. Contemporary European armies had large, state run stud farms to breed and train powerful warhorses, which could charge into the whirling musket balls and crashing cannons without stopping. Furthermore, they had a long tradition of training heavy cavalrymen to crash into the thick of the enemy formation. When enemy infantry would form square to hold against them, they had batteries of artillery on hand to devastate the concentrated target. I think a lot of the reason artillery caused so few casualties in the Civil War was due in part to inefficient organization, but also because they didn't have cavalry forcing enemy infantry into squares. Beyond the actual close engagement of the armies, this kind of cavalry, able to cut down enemy infantry with cold steel, was essential for breaking an enemy's morale, routing them, and pursuing the dregs, without which there could not be a decisive victory. In the Napoleonic Wars, cavalry and its accompanying artillery would viciously run down an enemy on the retreat, destroying their rearguards and scooping up prisoners. This didn't happen in the U.S. Civil War, largely because the training infrastructure necessary for making thousands of high quality cavalry troopers just didn't exist.