Slitherine

What is the best way to supply Japanese troops when invading from sea? Support ships? DD? or capture coastal towns? Dont move when landing?

Destroyers & cruisers, if you haven't got supply ships. Capture a town as soon as you can.

Note that cruisers can't enter shallow water, some coastlines may not have any deep hexes in the area needed.
I find destroyers best as you usually want to use your cruisers for proper naval battles.

Destroyers is the best bet since the bombardment is deadly hence a very good support unit

Yes. In coastal bombardment, destroyers at short range do the most damage. However, BBs and Cruisers cause a lot of damage to efficiency, and can fire much further in land, so they are also very useful when supporting an offensive.