There was certainly a desire from the German military and political leaders to test their military in Spain. Not only were Franco's Nationalists seen as a potential fascist ally, but undermining the Republic in Spain would deprive France of a friendly border to their south. In addition, the Germans were able to secure favorable trade access to Spanish iron and other commodities. As such, military assistance had a number of benefits.

However, developing blitzkrieg wasn't really one of those benefits. Certainly the Luftwaffe gained a great deal of experience in Spain via the Condor Legion. They kept up a rotation system that allowed a good number of pilots to get a bit of experience in real combat conditions. They tested out different bombing techniques ranging from direct attacks on ground forces, interdiction attacks on logistical lines like roads and railroads, and strategic bombing such as the infamous bombing of Guernica. Their air doctrine was tested along with their planes. When the early biplanes were found lacking, the Bf-109 was introduced and was very successful. When early formations were found lacking, the Germans developed a looser system of flying with wingmen that proved to be quite successful. However, the Germans didn't have a large number of troops on the ground. Their observers did closely watch the various battles of the war and contributed to a good number of them, but they didn't have the forces available to really test blitzkrieg ideas. (I must note here that blitzkrieg is a problematic term, and was not widely used by the Germans themselves. It is a blanket term that has limited utility. However, it can be used as a catch-all for the new methods of warfare for WWII, so I think it's applicable here)

The group that did have the troops available to try out armored assaults into the enemy's rear with the support of air power were the Italians. The Italians contributed roughly fifty thousand to Germany's sixteen thousand troops, and had a large ground contingent. At the Battle of Guadalajara the Italian CTV (Corpo Truppe Volontarie) tried to use their armored vehicles to pierce the Republican defenses. There were a number of issues that caused them to fail. Perhaps the biggest issue is that Franco never committed the troops he had promised to attack from another direction, leaving the Republicans able to reconfigure their defenses. In addition, most of the Italian armor were tankettes—smaller tracked vehicles that generally lacked a turret and a large gun. Instead these small vehicles had a machine gun and light armor. This left them potent compared to a Republican rifleman, but not nearly as formidable as the tanks that were used in the armored attacks in Poland, France, and Russia (Even the Panzer I and Panzer II outclassed the Italian tankettes). Further, the weather worsened, which reduced the ability of troops and vehicles to move effectively. The few Republican tanks (mostly early Russian designs that were nearly obsolete, but still superior to the Italian designs in that they featured a turret and a large gun that could penetrate enemy armor) were also critical to providing opposition. Finally, the Italians themselves weren’t sufficiently aggressive once they’d made a breakthrough. Instead, they awaited Franco’s promised assault in vain instead of pursuing the advantage they had early on in the fighting.

Different observers of the battle took away different lessons from it. The Italians were dismayed at their failure, but still had faith that these tactics would work in the future. The French and British largely took Guadalajara as confirmation that armor could not operate independently without infantry and artillery support. On the other hand, the Germans took away the lesson that the attack should have been more aggressive and could have been successful if it had been sufficiently supported.

So overall the Germans did learn from their participation in the Spanish Civil War, but didn’t directly test out the ideas that would become known as blitzkrieg.

For further reading, Antony Beevor’s Battle for Spain covers Guadalajara and the rest of the military aspects of the Spanish Civil War quite well. In addition, a declassified 1957 document from the US Air Force can be found in pdf format at the bottom of the linked page. It has two parts, and is an English translation of a German officer’s analysis of the war. As always, followup questions from OP and others are encouraged.