Defense depended on the terrain and the units involved
The initial plan for Fall Blau was to surround and destroy Soviet armies in the southern sector of the front, just like they did the previous year, and only then to move towards their main objective (i.e. Caucasus oil) and secondary objectives (Stalingrad, Black Sea ports, possibly Astrakhan etc ..). Due to reasons that are beyond the scope of this answer, this did not happen. The Soviets began a disorganized retreat, with huge losses, but were not completely destroyed.
The situation in August/September of 1942 begun to get complicated for the Germans as Soviet resistance stiffened, and their own forces were getting stretched and away from their supply bases. On the coast, the Germans and Romanians were having a problem capturing Novorossiysk. They would eventually succeed on 11th of September, but could not go much farther (the front would remain static there). In the direction of Stalingrad, after the battle of Kalach, fighting now moved into city itself. This is a well-known topic, so we won't dwell on it; it is sufficient to say that this part of the front also became mostly static in September.
This brings us to a main point, i.e. the drive towards oil fields (Grozny), and the simultaneous attempt to cross the Caucasus mountains and cut-off Soviet forces on the coast (thus securing their right flank). Soviet historiography considers the latter to have begun on the 9th of August 1942 with the fall of the town of Aramvir. At that moment, the Germans were breaking through into the northern Caucasus plains, and there was a danger they would turn southwards through the mountains towards the port of Tuapse. The Tuapse defensive operation, fought by elements of Transcaucasian front vs elements of Army Group A, is little known to the Western public. Without going into too many details, the German advance in this sector was slow and bloody (as would be expected in mountain warfare) before being completely stopped in November.
Further south-east, the Germans attempted to reach the Black Sea coast at Sukhumi, and this is the place where the famous capture of Mount Elbrus happened. It should be noted at this point Soviet forces were weaker than the were at Tuapse, simply because they did not expect a German offensive in this direction (worse terrain). In this part of the front (High Caucasus) snow begins sometimes even at the end of August, roads are barely mud tracks, so the whole expedition was of little military value. In any case, the Soviets managed to block this direction with little effort, since the Germans could not push more then few battalions of mountain troops in that direction.
Near Grozny, Soviets built their defenses on river Terek as a natural obstacle to the German drive towards Chechnya. The Germans crossed the river near Mozdok, thus beginning a battle known as the Mozdok-Maglobeg defensive operation. In the course of the battle that lasted whole of September, Soviets tried to reduce thr German bridgehead, while the Germans were attempting to expand it. The Germans moved their 5th SS Wiking division from around Tuapse, and finally managed to capture town of Maglobek which is about 30-40 km from the river, but at that point their efforts were being contained.
The final German effort came in late October and early November of 1942, at first having surprisingly good success due to unexpected direction (from Nalchik towards Vladikavkaz, then Ordzhonikidze), and became known as Nalchik-Ordzhonikidze Defensive Operation. Surprise came mostly because the Germans secretly regrouped 1st Panzer Army. However, by this point, the Soviets had enough reserves available to contain this offensive (in fact being readied for the Soviets' own offensive) before the Germans could achieve their strategic goals. It should be noted that in this particular operation, VVS actually started to appear over the battlefield, and managed to successfully contest Luftwaffe air superiority for practically the first time since the German offensive in the South started. Also, Soviet anti-tank capabilities (anti-tank guns above everything else) were increased, and in some sectors the Germans simply could not push trough with their panzers like they used to.
Overall, as the Germans expanded from the initial starting point of Fall Blau deep into Soviet territory, their troop concentration became diminished, especially after sustaining losses and moving away from their logistic points. Mountainous and overall bad terrain certainly did slow them down. Despite occasional successes of German or even Romanian mountain troops, they were not able to maneuver large formations and win decisive battles in such terrain. Even in relatively flatter terrain, obstacles like rivers slowed and sometimes completely stopped them. It should be noted that bad weather certainly played its part, as the Luftwaffe was unable to give full support (unlike for example in the summer months). Thus, blitzkrieg gradually gave way to a war of attrition which was ultimately more favorable to the Soviets.