Great question!
I would completely disagree with the other posted comment that it taught the Generals about the 'uselessness' of conscript soldiers. This was the before / opening battles of Britain's New Army, a fully conscript army, which would be put into full use on the front and was actually planned as a way to hold off the Germans long enough for said conscript army to be put into place. I would also disagree with the popular notion that it taught generals that they could not rely on artillery barrages to flatten enemy positions to be "walked over" by your infantry without resistance because the circumstances around it were not conducive to making widespread operational changes (which I'll explain later). That instance would be Passchendaele in 1917. What is very distinctive here is that it didn't teach the Generals that lesson but it is the first instance where it was attempted and we can see it failing with hindsight. That is why it is so commonly cited as an important battle.
Honestly, and I know some people would disagree with me here, the Somme didn't really teach any overwhelming 'things' to anyone because it did precisely the job it was meant to do. Let me explain:
It's December 1915 and the Entente leaders meet with Joseph Joffre to plan a massive offensive. Britain had been spending the past year building a massive conscript army, titled "Kitchner's Army" or "The New Army" and were already being deployed in the front. However there were millions of more men and they needed to be equipped, trained, and deployed. Further and perhaps most importantly, with Italy in the war, they would be capable of performing a 3 front assault on the Central Powers and squeeze them under the weight of the combined powers. Notably for our story, the offensive would be taking place in Picardy where the French and British lines met with the French meant to take the overwhelming brunt of the duties for aforementioned reasons.
Basically, their plan went to shit almost immediately. The Germans would launch an offensive against Verdun in February of 1916 when the Somme offensive was meant to begin in August of the same year. This would be the bloodiest French engagement in their history and would claim millions of lives. The German plan was genius -- take the Côtes de Meuse, the Meuse Heights, and force the French into a series of bloody but impossible counter-attacks and offensives to which the Germans would cut them down. The French would throw their full force into this offensive and were no longer capable of carrying the other offensive -- an offensive the British frankly were not prepared to do on their own. I don't have the numbers in front of me but I know the French would only contribute 1/4th of the amount of men they initially promised to when it came down to it.
To throw gas onto the fire, the commander of the French Army, Joseph Joffre, was badgering Haig to perform the offensive early. The fact remained that if Verdun was going to continue the way it was going, the Western Front was lost. She was losing far too much men and as Joffre put it himself, (and I'm paraphrasing here), if Haig didn't perform the offensive a month early like he did there wouldn't have been a French army left in August when it was planned to start.
So, we have a massively undermanned army attacking arguably the most fortified portion of the Western Front severely ahead of schedule. Of course it leads to disaster on a casualty scale. However, it did precisely what it was meant to do. The Somme saved the war. The Germans had to redirect swathes of men from Verdun to repel the British which gave the French room to breath and fight back and rebuild. It gave the British time to fully integrate Kitchner's Army into the front and prepare the front for 1917 offensives, notably the Third Battle of Ypres (Passchendaele). So in many ways there was no reason to critically assess the performance of the army as it did the job it was intended to do under the circumstances.
I think if any army learned the most from this battle it would be the Germans -- that they could not survive another Somme. They were lucky this time as it was the trying ground of a conscript army that was just being brought to the front but they frankly did not have the manpower to sustain another offensive like that. It would lead to the immediate construction of the Siegfriedstellung, the Hindenburg Line, a massive defensive fortification line meant to allow the Germans to retreat to a preplanned defensive position in the case of an Entente breakthrough. Basically if the Entente tried another offensive on the scale of the Somme they would win it initially but couldn't dare of attacking the Hindenburg Line. At least, that was the idea. It was the battle in the middle of the year in the middle of the war and it was arguably the tipping point the Germans realized they were outnumbered and outgunned by a combined British and French force that was now going into full gear mobilization.
Now that's not say the British or French were idiots -- they certainly began to see tanks as a useful tool which would be exploited in the Hundred Days Offensive two years later. They would have the idea planted in their head about the futility of relying entirely on artillery barrages for major offensives which would be solidified in Passchendaele the following year. However, the French didn't quite 'learn' anything as they were busy fighting for their lives in Verdun and for the British it did exactly what it was meant to do. A disaster in terms of loss of human life, no doubt, but it was for all intents and purposes a strategical success by the British. The Entente would take more land in that offensive since the counter-offensive after the Marne two years prior where progress was at last measured by kilometers rather than meters. It seized initiative away from the Germans which they had gained at Verdun and they would only temporarily regain it in early 1918, only to be stripped away again by yet another counter-offensive -- that time by the French and Americans though. Perhaps most importantly, while the British lost a significant and appalling number of men the Germans lost even more.
Notes:
The First Day on the Somme: 1916 by Martin Middlebrook
Tommy: The British Soldier on the Western Front by Richard Holmes
The First World War: Germany & Austria Hungary 1914-1918 by Holger Herwig