I can't answer for every aspect of the campaign, but I can give some specifics on the air campaign that largely parallel developments on the ground. In general, the trend was that the VVS (Soviet Air Force) was caught with its pants down in the early stages of a major rearmament to more competitive modern types. Training was lacking across the board both thanks to the infamous purges as well as the rapid transition to new types, and organization on almost every level was extremely poor.
To start off, let's look at the VVS inventory on the five western military districts on June 1, 1941:
| Total | Unserviceable | |
|---|---|---|
| All Types | 8,342 | 1,173 |
| Modern Types (MiG-1, MiG-3, LaGG-3, Yak-1, Pe-2, Il-2, Yak-2, Yak-4, TB-7) | 1,367 | 102 |
From the start, we see that the VVS was primarily made up of obsolete and obsolescent types, and it had a large inventory of unserviceable aircraft. Exacerbating the issue was the piecemeal fashion in which modern types were being deployed - rather than replacing obsolete aircraft from one unit at a time, many units entered the war with a varied inventory of modern and obsolete aircraft. Already poor training was alarmingly poor for most of these modern types, with many pilots having never even fired the guns on their new fighters when war broke out. Aircrew statistics shed light on how bad things were - of 7,222 available aircrews, only 602 were trained for modern types. Of those, 6,385 were considered trained for simple daylight combat in good weather, while only 221 crews for modern types were up to this standard. When we move to more complex conditions, the situation drastically drops off. Only 1,514 crews were trained for combat in poor weather in daylight and 1,174 for good-weather night operations, and in both of those categories a whopping 4 crews were trained for modern types. For poor-weather night flying, only 68 crews were trained, none of which were trained for modern types.
However, looking at the Luftwaffe force they went up against, things don't seem as bad. A total of 2,790 aircraft were deployed for Operation Barbarossa, including:
Unfortunately for the VVS, the Luftwaffe had made extensive plans from the start to account for this massive disparity in aircraft. They had performed extensive reconnaissance in the weeks leading up to the war to map out VVS airfields, and they opened the invasion with a force of 500 bombers, 270 dive-bombers, and 480 fighters to strike 66 airfields along the front. Despite indications that war may be coming, the VVS had made no efforts to protect its aircraft from surprise attack. Most airfields had the aircraft parked in neat, tight rows along the runways. Only token resistance by lone fighters or antiaircraft guns was offered. Some units sortied bombers to counterattack, but these sorties were poorly organized and without escort, and they were annihilated without accomplishing anything. The result of the first day was catastrophic to the VVS - the Germans recorded over 1,200 Soviet aircraft destroyed - 800 of which were on the ground - while losing only 63 aircraft themselves on the first day. By the end of the first week of the war, the Soviets had lost some 4,000 aircraft.
Following this initial shock, the VVS's continued failures are largely ascribed to the aforementioned poor training, tactics, and organization. It's telling of the state of the VVS that a major point of success for Alexander Novikov, commander of the Leningrad District air forces and later leader of the entire VVS, was his decision to have all aircraft fly with radios (many crews removed them to save weight) and send bombers out with fighter escort. Things that are considered very basic tactics today like formation flying with fighters were almost completely lacking, and often times bomber escorts accomplished little, as they were easily peeled away from the force they were supposed to protect. Even as the VVS began to recover over Moscow, glaring deficiencies in the quality of the force shone through with things like the use of giant cloth arrows on the ground to direct aircraft to the front and the regular use of ramming by fighters intercepting bombers over Moscow. Poor organization of air forces as a whole continued for some time.
To me, the most glaring example early in the war of the poor organization of the VVS is the failure of the Demyansk and Kholm air blockade. Following the first Soviet winter counteroffensive, they had encircled large forces of men at Demyansk and Kholm. The Germans had resorted to supplying the forces via airlift (Demyansk) and airdrop (Kholm) until a counterattack could relieve them. From February to May 1942, over 100,000 German soldiers were encircled in the pocket and supplied purely via air, and the Soviet air force was unable to interdict these transports to starve out the pocket. The German forces were ultimately able to break out of the pockets and retreat at the end of May despite the best efforts of the Red Army and VVS.