Your posts are about 6" square and I would suggest a 12" square concrete pad.
30cm would be suitable depth. Where you cannot achieve that, double the width and put put mesh or crossed rebar in.
Unlike a fence, it will not have a strong sideways force, unless wind blows on the open front
For elegance, you can use a small wooden form so the top of the concrete, visible above the ground, is not much bigger than the post. You need the post to be above ground level to reduce damp and rot. Your metal shoe will help. You can get steel studding to bolt down the shoe and press it into the fresh concrete before it sets, using the shoe as your guide. Dome the top of the concrete to throw off rain.
You could hammer rebar into the ground before pouring the concrete to give a bit of extra pullout resistance.
If you want, you can paint the shoe and the visible concrete to blend in with the timber stain. I use "Bitter Chocolste" masonry paint and a dark brown stain. The paint, and a metal preservative primer, applied before assembly, will also protect against rust. I am in a coastal location and use stainless nuts, washers and studding to resist salt spray.