The important thing with divisions is to know that you have two kinds of brigades: Combat brigades and support brigades. These are easy to tell apart, as all combat brigades have 3,000 soldiers and all support brigades have 1,000. Why it's important to know this is because support brigades do not take any width in combat, which can give you a considerable advantage in firepower. The composition I use the most is three combat brigades and two support brigades (needs Superior Firepower tech, easily gained in mid-1939).
Now, the first consideration when building a division is terrain and mission. This is most important with tanks, marines and mountaineers. You see, support brigades also influence the division's terrain penalties and mission efficiencies. This means that adding anything except engineers to Marine or Mountain divisions is counter-productive. However, I've found that two engineer brigades are a good supplement for three Marine/Mountain brigades because of the additional terrain bonuses, river-crossing bonus, movement bonus, defense bonus, but most of all the attack bonus against Fortifications. Even a level 1 Fort can be a pain in the ass.
With tanks, the consideration is that tanks suck anywhere except easy terrain, and river-crossing is immensily hard. To counteract this, two measures are needed: Combined Arms bonus and Engineer bonuses. Engineers make tanks considerably more useful in the tougher Asian and Russian terrain types, able to cross rivers with respectable ability, and give Fort attack bonus. Motorized brigade in addition to the three Armour brigades (Armour is the best tank type) eats up front, but there are advantages (in my opinion, decisive advantages) to using them instead of the more powerful Mechanized brigades: 1) Mechanized brigades are expensive as fuck, 2) Motorized consumes less supplies and oil, 3) You can begin constructing Motorized brigades considerably earlier, and 4) Mechanized can stab you in the back late game and get too low Softness which negates Combined Arms bonus.
Because you'll be fighting mostly in low infrastructure jungles, your best bet is to focus on Marine divisions of 3 Marine Brigades and 2 Engineer brigades. You don't even need to really bother with tanks or infantry, because Overlord is completely unnecessary in HoI3, since the Soviets will utterly crush Germany before they're even in Kiev (you won't believe the amount of modding tricks I've been trying to do to fix this).
Otherwise, you won't need Infantry divisions until you reach the Home Islands and Chinese mainland. Armour isn't very useful in Japan, as Japan is very mountainous and has lots of urban provinces. Southern China has a lot of Plains though, so a Corps or two of tanks can be very useful. The most important thing you need for an invasion after the fairly light land fighting in Pacific is a massive transport fleet, no need for escorts, because that's what your actual fleets do. Just stick some Old Fart Guard admiral with low skill at the helm of the 40 or so transport flotillas.
For naval warfare (I'm gonna post in this regard anyway), in my experience the strongest navy possible relies on Light Cruisers, Battlecruisers, Super-Heavy Battleships and Carriers organized into fleets where the Positioning penalty is less than 20%. Having two carriers each fleet is a good idea.
Destroyers are worthless, IMO, because they sink like rocks. The Light Cruiser provides better protection at only marginally higher cost and longer build time, while having immensily larger staying power. This is important, because the less you need to waste time building replacements for your screens, the more time you can spend sinking enemy fleets. Naval warfare in HoI3 is more than anything about attrition, as replacing ships takes much, much more time than repairing them.
Super-Heavy Battleships are better at everything than Battleships (may not be the case in vanilla, but it is in ICE and later updates where they upgrade from techs), for relatively slightly longer construction time. They're also goddamn indestructible and shoot at prepostrous ranges, making them ideal for protecting carriers.
Battlecruisers have smaller hullsize than Battleships, noticiably cheaper and quicker to build and greatly faster but aren't critically weaker compared to Battleships. This makes two Battlecruisers supporting the Super-Heavy Battleship an ideal solution for the Carrier fleet.
Having a large number of Interceptors is a good idea, because Naval Bombers and Tactical Bombers can be dangerous to your fleets, and the enemy might have Interceptors of their own which could neutralize your CAGs.
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