From: High Point, North Carolina
I'm by know means an expert here, but I will give you my thoughts on some of your questions.
Yes you should have a mixture of marines, infantry, armor, artillery and engineers. Get the high assault value units on land on the first wave and the others quickly after that. Be sure to get in engineer units that also have an assualt value, as they assist in knocking down enemy fortifications. For this scenario, theres a special HQ that can be loaded on a special transport that does add strength to an island invasion, but I can't recall the proper names right now. This HQ stays on the ship in the coastal hex, don't unload onshore.
Air Combat task forces should have a fast battleship or two in them, in my opinion. I like to group three CVs in one TF and use CVLs in separate TFs to guard the perimeter of the battle--out of reach from land bases enemy aircraft or at least a long flight for them. There are many schools of thought on this topic, but this has worked well for me against the computer.
You are using the transports and landing craft very well, no problems. There's a chart in the manual that lists exactly what each type of transport is capable of hauling and how much. I have this page printed out and refer to it regularly. Remember though, early in a long game you have to use whatever is at your disposal, even if it isn't ideal according to the chart.
It sounds like your ASW TF returned to base because it was in retirement mode rather than patrol. Any TF in retirement mode will go to the hex you selected and then return to home base. Use patrol to get a TF to stay in a hex or move it from there at a later time.
It sounds to me like you're ready to play a long campaign. Finish this scenario if you're having fun, but there's not much left for you to learn here. The opening of the long games play nothing like this one because you won't have anything close to this level of lift or firepower for two years in game time.
Best to you in your game, WITP is a blast!
USS Terry (DD-513) — Battle of the Atlantic, Solomon and Marianas Islands campaigns and the Battle of Iwo Jima