I'd use 1/2 inch MDF for the sides. For a small cabinet, 1/2 seems better to me.
For all the battens and center panels, I'd use 3/4 inch plywood. It's strong and takes a screw real well.
For the CP, I'd also use the 1/2 MDF. You can shape it easily, and glue holds MDF just fine.
I think some hardware stores, Home Depot, and maybe Ace sell sheet material cut in smaller sections (2x2 or 4x4). Not 100% sure... it's been a while since I've looked at that sort of thing. Anyhow, it might keep you from having to buy full sheets.