Goodmongo has good advice! Also, the Italians usually can hold until the DAK arrives. Oh, what I do, is make sure to have 3 Italian armies leading the assault; the armies hold up much better than corps.
Another part, if supply is a huge issue (and this is my favorite tactical use of German air), move all the medium and TAC aircraft to Sicily as soon as they aren't needed in France. That should give you 5 bombers, and it will take 4 turns (maybe 5) to fly them down to the boot toe. Operate an HQ there too. Bomb Malta clean, and use an Airborne to take the place when clear. Use the Italian sub to scout the island (make sure to move it away from Malta after you bomb, and keep in silent mode). Yes, it will probably take away the air from U.S.S.R. in the opening offensive, as I move the air through the Balkans to help (especially Greece).
Part Two: (Fire in the East)
Wow, this is nothing but a meat grinder. You should have 7 tank units, and 3 mechanized units to open the offensive. The tanks should be at least Tech 2, and the Mech should be at Inf 2. I break it up with 2 tanks north, 3 center, and 2 south. The 3 Mech have all been going center. The most important thing is to have as many HQs as possible. I usually have 5 or 6 to open the war. Try to kill units when their supply is 4 or lower. Even if you have to wait one extra turn, it is worth it to make Uncle Joe pay full price to bring those units back to life.
This is Terif advice: Use forced march to move them Germans west, then east as much as possible, and Operate as little as possible. That will add up to quite a few units in a short amount of time.
Terif btw, is one of the firstest and bestest SC players, evah! Terif, if you are still around /bow. You are missed! He had a page of advice, and it was for the first version, but it still applies. Can't find it, but it might be out there somewhere.
Long term advice: buy one piece of tech every turn until you hit cap. Mixing it up between tech and units is so important. Of course, there are some times to change it, but if you follow this idea, this mix will keep you competitive in most games. I like to save up extra MPPs from September 1941 through the end of the year. Reason being to help recover from the Severe Winter in Russia, and to get the 3 Heavy tanks as soon as possible. So, having an extra thousand MPP sitting there in October is very useful.