What was air combat like for the Soviets during WW2?

Not very well initially, needless to say. The bulk of the soviet air force was destroyed on the ground, and the air force was in the midst of a modernization. The I-16's and I-15's, still common, were completely outclassed by German fighters.

Other, more modern Soviet fighter aircraft around at the beginning of Barbarossa were the Lagg-3, the Mig-3, and the Yak-1.

The Lagg-3 proved heavy and unmaneuverable, and was phased out from first-line air combat. Production aircraft often performed much worse than the initial prototypes because of mistakes with the wooden construction. Pilots often made the joke that the Lagg was a "varnished flaming flying coffin," or the equivalent in Russian, using the letters LAGG as an acronym. However, the Lagg-3 did see some service up until 1943 or so in ground-attack squadrons. It also did serve as the foundation for the very successful La-5 and La-7 fighters later in the war, but in July 1941 the Lagg-3 was no good.

The Mig-3 was probably the most common front-line newer-generation fighter at the start of Barbarossa. Pilot's views of this type were mixed- it was originially designed as a high altitude interceptor and featured exceptional performance at altitude, matching or beating the Bf-109F aircraft that were the mainstay of the Luftwaffe in the summer of 1941. However, these advantages pretty much disappeared at lower altitudes, where the German fighters had the edge. High-altitude air combat on the Eastern Front did not occur very often (large/dispersed armies and long distances meant that most air operations had ground support in mind and most operations were performed at a much lower altitude than on the western front- this is both true of the Soviet Air Force (VVS) and Luftwaffe). Because of this, the Mig-3 was often misused, and generally disliked. However, it was an effective aircraft when used the right way. Pokryshkin, the 3rd (?) highest scoring of the VVS during the war, like the type.

The Yak-1 filled pretty much the opposite role as the Mig-3, and served in approximately equal numbers in June 1941. It was a maneuverable low-altitude fighter. In the hands of a good pilot it didn't have too much of a problem against German fighters at the altitude of most VVS combat operations. It was popular with pilots, and served as the foundation for a series of successful developments (Yak-9, Yak-3, Yak-7), which formed the principal VVS fighters of the war.

So, long story short, the soviet air force was outclassed at the beginning of Barbarossa, and losses were heavy. Keep in mind that the above only focuses on fighter development, and that soviet bombers and ground-attack aircraft suffered even more appalling losses during the first few years of the war. There was a reason that many German aces achieved extraordinary numbers of kills on the Eastern Front- several breaking 300. Things like poor pilot training, inferior aircraft, masses of unprotected ground attack aircraft, and the like held the VVS back for several years. However, after around late 1943 or so, because of the new developments of the above fighters together with German attrition, the air war did start to become more even.