I believe that there are several previous debates answers that could be turned up if we wanted to attack the premise that the ACW is a trial run for WWI. Let's put that aside and look at Europe itself between 1865-1914.
The main driver of military change in Europe between 1866-1914 was because of the Prussian (as they became the Germans) general staff welding the technological innovations of the mid 19th century into a coherent strategic, operational and tactical doctrine. This forced France and other European powers into rapidly evolving their military doctrine.
Prussia moved from a third class military power in 1865 into a dominant first class power in 1871 by winning two wars, the Austrio-Prussian war and the 1870 Franco-Prussian war. Moltke gets a lot of the credit for this, but many Prussian officers and governmental figures deserve notice. In any case, it was the Prussians and their rivalry with France that sparked a technological arms race and a rapid and incontrovertible increase in firepower. The Prussian general staff crafted the mobilization and reserve schemes that made it possible for armies of hundreds of thousands and then millions to be formed within weeks or months. The Prussian and German general staff created operational schemes of maneuver that utilized railroad pre-planing in meticulous detail. Furthermore they realized that armies of hundreds of thousands of men could not all march or be supplied on 2-3 roads but would have to spread out across military frontiers in order to maintain mobility.
This idea of dispersion is probably the biggest change to operational strategy that accompanied the period. A late ACW battle like The Wilderness in 1864 or a European battle like the 1859 Battle of Solferino took up only a few dozen square miles. Hundreds of thousands would be concentrated onto relatively small battlefields that were controlled by a single general with a few mounted messengers for communication. By 1870, the Franco-Prussian war would see armies dispersed across the frontier and coordinated across hundreds of miles by telegraph and any modern means of communication available. The initial actions of WWI in the west were fought in more or less the same territory, except for the swing through the low countries broadened the war outside the French and German border.
The only thing that prevented the Franco-Prussian war from being a "trial run for WWI" was that it was mercifully brief. The Germans crossed the frontiers in August and surrounded Paris in mid September. The French learned many lessons from 1870, and imitated the German general staff in their planning methods. When the 1914 conflict emerged, they were better prepared and the resulting conflict shared the grueling length and indecisiveness of the ACW.
Sources:
Barry, Quintin "The Road to Koniggratz Helmuth von Moltke and the Austro-Prussian War 1866"
Barry, Quintin "The Franco-Prussian War 1870-71"
Nosworthy, Brent "The Bloody Crucible of Courage: Fighting Methods and Combat Experience of the Civil War"
John Terraine, in his polemic work The Smoke and the Fire groups the ACW, and the two world wars together as comparators but because these were industrial wars requiring a total commitment from belligerents rather than similarities with regards to technology and tactics.
Its a bit of a difficult question to address because of two factors:
Each of the belligerents in WWI had differing doctrines, while WWI itself saw such a rapid evolution in tactics and technology, and was fought in so many different theatres necessitating differing doctrines, that it is impossible to regard it as a single event from that point of view.
The claim that the ACW was in some ways a trial run for the Great War is itself somewhat suspect as the use of trenches in ancient and vastly predates both wars, and I am not aware of any actual chemical weapon deployments in the ACW, though I stand to be corrected on this.
The principle difference between the ACW and WWI was the quantum leap in the lethality of weaponry.
Consider for example the evolution of the personal weapon from the ACW to WWI. In the ACW, the standard personal weapon was the muzzle-loading, black powder Minie rifle. By WWI, the personal weapon had developed several generations, from muzzle loading Minie rifles, to Chassepot cartridge riles, to single shot brass cartridge riles like the Martini Henry, and finally to the magazine fed rifle firing sub-bore spitzer rounds at high velocity using smokeless powder.
Artillery had advanced almost beyond recognition, from muzzle loading black powder cannons to rapid fire recuperating breach loaders. A battery of French Soizante Quinze guns could saturate a 4000 square meter area with 17,000 shrapnel balls every minute. Direct fire, although still practised in 1914, was increasingly recognised as unsurvivable and being replaced by indirect fire, especially as a result of experiences in the Russo-Japanese war.
Although Gatlings were used in the ACW, these were arguably not true machine guns, and like the mitrailluse were more properly artillery pieces than tactical infantry support weapons. By WWI, we not only see the proliferation of Maxim guns and their derivatives, but also true squad automatic weapons like the Lewis gun.
Finally, while some cavalry arms had barely developed, some, notably the British, had evolved a model of cavalry being able to fight dismounted with the same proficiency as infantry, but also able to carry out traditional ‘cold steel’ shock charges.
The overall effect of this increase in lethality was that the line tactics that dominated warfare for centuries was simply no longer viable.
Infantry had to manoeuvre in dispersed skirmishing formations, artillery had to fire “off the map”, and cavalry, the only means of pursuit and exploitation was effectively neutralised by the proliferation of barbed wire.
This meant that troops had to act with greater independence and less rigid command, while the battlefield would expand from relatively small area to vast tracts of countryside, quite outside of the ability of one General to personally survey and direct. Waterloo was fought on a frontage of about 3 miles, while the First Day of the Somme saw action across 22 miles of frontage.
The problems of command, control and coordination were never really solved during the war – that would have to wait for the advent of portable radios.
Source
John Terraine – The Smoke and The Fire
Strong & Marble – Artillery in the Great War
Todman, Sheffield et al. – Command and Control on the Western Front
One of the popular misconceptions about the development of combat tactics in the run up to WW1 is that all the tactical developments of the ACW and the Franco-Prussian wars were forgotten just after those conflicts. This is, in fact, not the case. The stalemate entrenchment tactics of the age of the rifle were still thriving up through the battlefield phase of the Boer War. In one of the most interesting institutional and intellectual tales of history, the doctrines of the armies of Europe actually became substantially less well suited to fight WW1 in the 10 years running up to it.
As has often been pointed out, in its last two years the ACW developed into a war of entrenchment and flanking marches as it became clear that the firepower of rifled muskets had made direct assaults foolish. This was noticed in Europe, and one of the most dramatic changes in European armies in the 1860's was the near universal adoption of standard issue entrenching tools and the new expectation that all soldiers were expected dig routinely, not just specialized units.
Helmuth von Moltke was one of the most brilliant generals of his age, and he grasped clearly that these new developments were inclined to produce stalemate. That is why, when he commanded in the Wars of German Unification, he focused on rapid overwhelming concentration of force using rail transport and detailed mobilization schedules. But even in the largest of these wars, the Franco-Prussian war, in the largest battles the Germans proved incapable of forcing their way into French positions by direct assaults. All the greatest victories were built on outflanking and encircling smaller, slower-moving forces. And after initial stunning success even the Prussian war machine was drawn into the rifle stalemate around Paris as tiny forces proved capable of resisting larger ones through firepower.
The period of 1872-1913 was one of unusual peace within Europe. And most of the wars involving Europeans involved colonial campaigns against much more poorly equipped enemies. There were other disturbing signs of the deepening stalemate rifles could produce. At the Siege of Plevna (1877), Europe was shocked by the horrific casualties the Turks, armed with Winchester Repeating Rifles, were able to inflict on much larger Russian forces. This sent Europe scrambling to deploy magazine-fed rifles. But, ignoring a string of conflicts in the Balkans involving forces of mixed regularity that was poorly understood by the outside world, the next major shock came during the field army phase of the Boer War (1899-1900). This was the first time major armies fought using magazine-fed rifles using smokeless powder. The Boers fought from trench networks, and were able to stalemate and even nearly destroy seasoned, professional British armies sent to fight them. The British were only able to prevail by using larger forces and flanking movements. The lessons of this conflict were somewhat dulled by the long and politically costly guerrilla war phase (1900-1902) that followed and distracted attention from the first part. But generally speaking, the broad perception that warfare was built on digging and flanking was fairly set in the beliefs of European armed forces by the beginning of the 20th Century.
The critical and shocking turn in this tale comes in the Russo-Japanese War (1904-1905), which was the first conflict noted for the widespread deployment of Maxim machine guns. During this conflict, the Japanese (who had not really yet experienced the rifle stalemate first hand) adopted a policy of frontal assaults on Russian trenches and machine guns. This was where the legendary courage of the modern Japanese soldier began to display itself, and the Japanese were able to win several "victories" by carpeting the fields in their dead. In retrospect, it seems many of the major engagements were settled by retreat by unsteady Russian commanders (reminiscent of the ACW), but that was not the lesson taken by military observers at the time. From first hand reports of this conflict by professionals there began to grow a narrative in military thinking that machine guns could be defeated by riflemen with enough guts. This idea seductively appealed to the dime novel romanticism of the era, and quickly started to spread among military thinkers.
This fad thinking, that the era of entrenchment and flanking could be ended by guts (or "cran" as the French called it) infected most militaries to some degree, but the French caught it especially bad. In the 1870's and 1880's several countries were experimenting with "trowel" bayonets: a wide bayonet that could serve as a shovel. By 1910 the bayonets of most armies were getting longer, thinner, and more sword-like so that "our boys will have more reach than the enemy". In 1908 the British reauthorized the use of the cavalry lance. In one of the clearest examples of this process, the French reintroduced the cavalry saber. The US reintroduced the cavalry sword in 1906. George Patton convinced the US Army redesign the sword in elongated French style in 1913 to improve reach. Patton was then put on exchange to France in 1913 to study fencing with the French Army. He graduated in June of 1915 because the school was still functioning. It had not been shut down despite nine months of experience of fighting WW1. In keeping with their philosophy of using cran as the main weapon, the French engaged in extraordinarily insane attacks through 1914 and much of 1915, managing to suffer more casualties than the Germans despite ostensibly being on the defensive for much of this time. (If you want a great example of the French tactics during this time, look up August 22, 1914 during the Battle of Frontiers.)
The strangest thing about military doctrine in this 50 year span running up to WW1 is that it took a long while to get worse. And the reasons it became worse seem to only have a little bit to do with military experience. Much of it is to do with how people from that time period interacted with the world and formed their beliefs. I'm not sure you could fully understand it without studying not just military doctrine, but also art and literature and the prevailing political narratives. But that is for another post.
An excellent source on the changes in doctrine in the run up to WW1, I recommend GJ Meyer's The World Undone.