The side that is advancing (the Axis early on) have much more problems with their supply than the side that is defending its territory. So in the early going with the Soviets, I keep pretty much everybody on the map who has any conceivable chance of being useful. I send air units with morale < 40 back to NR at the beginning of my turn. You should keep an eye on fatigue, but it is unlikely for Soviet air units to have fatigue problems in the early going because there are just so many of them, and their morale is weak, meaning they will be in NR pretty frequently (where they will lose fatigue as well as gaining morale). Then I move air units if necessary (using the air kabuki method that Telemecus advises, though this is not essential for the USSR). I put all fighter-bombers on bombing missions except for the very highest experience guys, hopefully in modern fighters. Then I bomb every hex on the front except for those that have huge concentrations of flak (you should be able to figure out if the enemy has a big flak concentration with some air recon). I keep an eye on the Axis fighter response. If there are a lot of Axis fighters flying, I pound that area with as many missions as I can. When/if the numbers of fighters intercepting starts to go down, or if they are showing significant reductions in combat efficiency, then I hop on that fighter base with a big fighter sweep (f7 airbase bombing mission in which I deselect all the really good bombers). If fighters suddenly stop flying altogether, it means they have run out of fuel at that base, often a problem in the early going. In this case, send the bombers as you have a good chance to kill a bunch of fighters on the ground. If combat efficiency is declining, this means that they have very high fatigue levels, meaning you do the fighter sweep thing.
The rotation method that Telemecus discusses above is a way to address this latter problem, by ensuring a steady supply of fresh fighter squadrons in those front-line bases. The down side is that you don't have enough fighter numbers to cover the entire front efficiently. As the Soviet player, you should pay attention to what kind of fighters are intercepting you, or if there is a large region of the front where no interceptions are taking place. If all you see is Hungarian or Italian or Romanian fighters, go after them - they have lower experience levels that the German pilots and you should be able to knock them down pretty efficiently. And once they are gone, they aren't coming back nearly as quickly as the German ones. If there are no fighters at all, you can transfer most of your own fighters away, but leave a few because you might have the pleasure of drawing an interception or ground support mission that is unescorted. If you have a good fighter squadron, you can shoot down a dozen German bombers in one raid.
If you are playing v1.11.03, you need rear-area airbases where you can bring planes back from NR that are not yet ready to go to the front - some fatigue left, morale not yet up to scratch, or needing more experience. This is because of a bug in 1.11 where NR acts like an isolated air base and you can only gain one morale a turn after the first boost. In 1.12, this is still a useful move but not so important, since the morale gain bug has been corrected. But it is still useful to have some guys back in the rear with FB's set on fighter, so if the Germans try to rebuild morale and exp for level bomber units by random strategic bombing you can give them a bloody nose.