I'm installing a 320 amp single phase service. What size cable do I need to use? I'll be using copper. Also what size for 200 amp?

Assuming this is a residential service, the 320 amp will be minimum #400 MCM Copper with either THHN/THWN or XHHW insulation, the 200 amp will be #2/0 with either of the same insulations.

Assuming the 320 amp is the feeder to the meter, no equipment ground is needed. The grounding electrode conductor will be # 2 copper. The 200 amp is probably for the panelboards, and each feeder will need an equipment ground conductor of #6 copper. Be sure to separate, insulate and isolate the neutral from the equipment ground at each panel.