The First Horse Army, or konarmiya, of Semyon Budyonny (or Semën Budënny and similar) is one of the most famous units of the civil war. It was focussed around a mass of cavalry, but usually included some infantry units (and hence it is, strictly speaking, incorrect to translate it as 1st Cavalry Army).
As the cult of Stalin became more thoroughly pervasive, the role of the Konarmiya grew out of all proportion to reality because Stalin was very closely linked to it (being the RMS in Poland, for example). The Konarmiya commanders were some of the few senior officers to survive the purges, and from that point on it was safest to praise only them (or men like Chapayev or Frunze that died early).
Isaac Babel wrote the famous "Red Cavalry Stories", which had infuriated Budyonny because they portrayed the nature of his men during the Polish campaign all too accurately. Babel refused to rewrite the book as Stalin enforced Socialist Realism, and he paid for that with his life. In such an atmosphere, genuine historical investigation was not safe.
This was not just a passive exercise either. Stalin had Melikov's monograph "Heroic Defense of Tsaritsyn: 1918" rewritten to completely remove Zloba's Steel Division! (citation).
One area that is relatively easy to untangle is the idea that the First Horse Army was supported by a lot of technical arms. They appear on the orders of battle, but that does not mean they appeared very much in actual fighting. Budyonny was a simple man, and he had a simple plan – he would manoeuvre his army so as to find a weak spot in the enemy's line, then rush that in force, exploiting any penetration to the full. That meant avoiding the main rail lines and roads, which is where defence would be strongest, and not stopping to wait for the support to catch up. He really didn't get the idea of mixed technical arms, as would be exposed brutally in WWII.
Thus in Poland his Army was continually harassed by the Polish air force. Despite the oft-cited planes in his own forces, if they existed as working aircraft they were not present.
From time to time the First Horse Army included infantry. This was to hold the rear areas and flanks and to provide a screen when the cavalry needed to rest. It did not participate in the attack alongside the cavalry divisions.
The origins of the First Horse Army were during the unsuccessful Don Cossack offensives on Tsaritsyn. With insufficient infantry and no technical means to breach a fortified line, the Cossacks failed. The operations of the newly formed Soviet cavalry, while useful, was not the critical factor. But association with the defence of Tsaritsyn was the first step in the rise to fame.
The key operation that set Budyonny on the road to glory was the breach of the White lines at Voronezh, then Kastornoye. The break-through of the over-extended Markov Division and Cossacks took weeks, and was largely done by manouevre, as the minimal casualties show. Once through, however, the Horse Army was able to wreak havoc in the rear, and was a major reason for the utter collapse of the White front.
Its next operation was the difficult crossing of the Don. Initially, despite the enemy being in poor morale, things went quite badly. The Soviet strategy was not great, but this doesn't disguise the fact that the First Horse Army wasn't able to dislodge the Whites. Eventually, after a shift of positions, a hole was found, and the Soviets' bigger forces could be applied. Again the Whites collapsed utterly.
However, when the same thing was attempted at Kiev, the Poles reacted very differently. They fell back in good order, not panicked by having their lines penetrated. Communications issues and the poor discipline of the Horse Army meant the victory merely gained ground, rather than destroying their opponents. Later, when faced by the solid Polish lines at Zamosc, the Horse Army had no answer. And when it was their turn to retreat it was something of a rout.
I'm not suggesting that the First Horse Army was poor, but it gained a reputation for brilliance when faced with enemies who collapsed. When the enemy proved stubborn it turned out to be just another set of cavalry divisions – great at going forward, terrible at defending.
I've seen nothing in its history to suggest that the units in it were better than other Red Cavalry. The fear in the White ranks was what happened once it got into the rear of an army, not facing it frontally. Off the battlefield, it was famous for looting and pogroms, which is hardly ever mentioned in the Soviet history books but does not indicate a disciplined force. That said, Budenny did at least keep his focus on the enemy – unlike say Mamontov, who wasted a perfect opportunity when breaking through the enemy lines in mid-1919 to go on a pillaging raid.
The story of the other horse army is almost an inverse. Formed by a charismatic leader in the same area to fight the same foes, at more or less the same time, it was politically unsuitable for Stalin era Soviet history books. Built largely around Cossacks and with a commander with a tendency to criticise Bolshevik policies, it was simply not useful for propaganda purposes, and was largely written out of the war for that reason.
Yet it mostly performed as well as the First Horse Army, when commanded by Mironov. (Fighting Wrangel's initial breakout while under the command of the unstable and inexperienced Pavel Dybenko it did poorly, but that was a very poor appointment made for political reasons.)
Stephen Brown. The First Cavalry Army in the Russian Civil War PhD Thesis, University of Woollongong.
This includes highly detailed studies of the military aspects of the Army. Quite fascinating.
A Study of Bolshevik Cavalry in the French Revue de Cavalerie, July-August 1921.
In practice this actually studies the Konnarmiya during its Polish campaign, not Red Cavalry as a whole.
Semyon Budyonny, "The Path of Valor", Progress Publishers, 1972
The English version of Budyonny's already unreliable memoirs, leaving out all the reverses, and especially Poland. Nevertheless, you do see the sort of man Budyonny was – violent and not troubled much by deep thoughts.
"Who Took the Crimea?", 1973.
An article by four Soviet historians about the distortions in Budyonny's memoirs as they relate to late 1920. In particular how Budyonny deliberately tried to reduce the role of the 2nd Horse Army and elevate the role of the 1st Horse Army.
Sergei Starikov and Roy Medvedev, "Philip Mironov and the Russian Civil War", Knopf, 1978.
A biography of the commander of the 2nd Horse Army. Not particulary good on military detail, but shows how important it was. Again this is a Soviet era account of the period.
The "Eyewitnesses" section of the site has material from men who faced the Konnarmiya.