This will depend on many things. First of all we have a nasty weather bug, this will ensure there are always raining and heavy bad weather in all sea provinces. This will have great impact on the selection of your fleet compositions.
I have removed all the weather penalties (overcast and rain) for airplanes and ships until this is fixed so I can only speak of how I do my fleet composition based on that.
Carrier fleet
In a carrier fleet you want several Carriers (in 1.2 those CAGs are way more useful in greater numbers than before) as the core. The number would be between 2-5 depending on your resources. I would apply an equal number of other capital ships, preferably ships with speed to match the carriers. Some older BB are too slow. But if you don't care about speed but more about power then you could include the BB as well.
As my screen I like a mix of Destroyers and Light Cruisers, usually equal in numbers. Some say Destroyers lack the range, but I seldom find that is much of a problem. They are also great for finding and destroying submarines.
Battle fleet
If there are enemy carrier task forces around you need to make sure these are protected by air-power (Fighters and Naval bombers or land based CAGs).
Other than that they are just a fleet of whatever BB,BC and CA that you have and can spare for whatever missions you want them to complete. Such as Intercepting enemy fleets or giving support for a naval invasion somewhere.
I try to keep these fleets at about 18 ships. 8 capital and 10 screens.
Patrol Fleet
I run them in different sizes depending on the area and what I want them to perform. Usually 1xCl and 2xDD are my standard configuration on a defensive patrol mission. Sometimes I include a 1xCA,2xCL,2-3xDD if I want them to engage minor fleets that they encounter, such as weakly protected transport fleets or other weak patrol fleets. Otherwise I keep my intercept fleet in a port nearby.
Submarine Hunter Fleet
This need to be big enough to ward of weaker enemy battle fleets and to encounter any subs they find. One typical such can be (resources permitting) 1-2xCVL,1xBC,2xCA,4-5 CL, 8 DD.
This feet should be able to handle most Subs and enemy battle fleet that it encounter, though they will suffer against any serious enemy CV fleet.
I usually guard my smaller transport fleets with a CL or two while larger transport fleets gets a larger complement of ships. I never put larger ships than CA in transport fleets in general. I rather accompany such important missions with a real battle fleet or carrier fleet.
I only use submarines for convoy raiding, patrol and light fleet raids. Submarines can sink ships, but usually only during poor weather, at night or ones guarded with few destroyers.
I employ convoy raiders in 2-5 fleets, depending on the strength of the enemy convoys. The smaller the fleet the less chance they have to find them.
I send patrols in 3-6, depending in the aggressiveness I want them to have, but usually three strong are good enough for me.
For light strike duty I simply put a stack of six on the intercept mission and send three on patrol.
These are my thoughts on naval combat... though, there might be those with more experience on naval warfare than me.
