Ya, don't try what I did by bashing your head against a wall repeatedly by trying Nationalist China as your first campaign over and over until you started figuring things out...
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Rumania/Hungary I found worked quite well for my first couple of games. You are not hugely important for the entire war effort, though you still have your place. Further the relatively small territory and army are easier to manage. I would not recommend USA however, because a game where you are required to use all three military branches in tandem on multiple fronts at the same time makes for great difficulty.
I wouldn't advise the US as a first game, The Tandem Naval/Air game can be quite challenging to figure out in the beginning, when to build ships, when to build an airforce, how many infantry do I need? How many tanks? You basically start out with nothing (a decent, but aging Navy) but little in the ways of anything else. The perks: Large IC, decent Manpower, Later Entry into the War.
Germany is a good place to start, excellent Tech Teams, Awesome Generals, Good Base IC, Fantastic Ground Doctrine, Relatively weak early game opponents. Blitzkrieg tactics can cut through anybody with ease, just remember Tanks-Motorized Infantry (later Mechanized)- Stukas. Russia isn't hard, as long as you don't repeat Germany's historical mistakes.
Disadvantages: Terrible Navy (but if you conquer Russia, trigger Bitter Peace Event) a Sealion is a much easier feat to pull, you can do it early game, but it takes some...finesse.
Using Tanks-Motorized/Mechanized will basically cripple your oil supplies and TC before you get far into Russia. Better if just to use them as Armored Spearhead with main army Infantry divisions following their progress then have an army composed mostly of Armored divisions.
Yeah, I should have clarified a bit. Tanks-Motorized-CAS for the 'Armored Fist' and lots of Infantry to tighten the gap. A stack of 2-3 Tanks + 3-6 Motorized/Mech usually does the trick for me (This was a major problem I had with Germany the first few times playing), and you only need 2-3 Stacks of these to make short work of Enemies. I usually rush for Baku and risk the potential TC Hit, however cutting off Russia's Oil Supply, and gaining the 20 Oil for myself generally allows you to open up a new front in the North for a quick rush to Leningrad/Moscow. Especially with the killer CAS-TAC Combo, you can make short work of the Soviets even if the majority of your holding forces are Inf. Of course this wouldn't work very well against a Human player, and on Very Hard I found this tactic to be useless, but on Medium/Hard I haven't had much of a problem.
See, that's the beauty of playing the Soviet Behemoth. While German spearheads consist of 9 Armored divisions. In my game, my Armored spearhead and so overwhelming they break anything.
It usually consists of using a Panzer Leader Field Marshal, with 12 Armored Divisions under his command, followed by two Generals using 9 Motorized Divisions, followed by a gigantic wave of Infantry divisions. All this while some 50 CAS squadrons are on the air obliterating everyone in its path.
As to your strategy, you would have to hold the North Crimean path all the way to Astrakhan solidly and with decent depth from the Black Sea to allow for the Oil Production in Baku to rise to something significant. Otherwise the Soviets will just counter-offensive somewhere near the Sea of Azov, and your >10 divisions holding the Caucusus are :daisy: up, unless you have Air Supply on, which provided you can defend the Caucusus appropriately, it allows you to hold your ground long enough to bring down reinforcements to try and link up with the Caucusus again.
In my last Germany game, where I tried to play peacefully (Didn't go for war in the "Danzig and War" event), since the USSR declared war on me, they didn't benefit from the "Moving Industry to the Interior" events. After I had duly dealt with France (Which had declared war on me before - By dealing with them, I was blitzed the Maginot Line and dug in), I made a push Northwards, which after the success of that, the Soviet IC crippled, and the rest of the campaign was fairly easy (I made another push through the South and managed to encircle like 80 divisions around Lipetsk - Before that, the Soviet Union had a giant army advantage over me. Like 150 divisions of advantage).
In normal games I usually push both South to Stalingrad and North to Moscow/Leningrad first.