I know many do it, but I've heard from knowledgeable people I trust that it's better not to use concrete to set wood posts, as it'll eventually rot them out. Of course, if the posts you and your dad set in concrete feel solid, i.e., they don't wobble, at all, there's no sense digging them up. Just telling you for future reference.
When setting a wood post in the ground, I always try to dig the hole as small as possible, usually just one shovel's width in diameter. I put a few inches of crushed stone aka gravel into the bottom of the hole to make sure the post isn't sitting directly in mud. Then, I refill the hole with a combination of the soil that came out of it, alternating with spadefuls of gravel and sand, tamping everything down, as best I can, while I'm refilling the hole. When you're done, the post should be solid as a rock. You shouldn't be able to move it, at all.
As far as the bracing goes, your idea of putting two diagonal braces seems reasonable to me. You'll want to do it exactly as you drew the diagram. Start low on the right post, closest to the house, and go upwards to the post the gate is attached to. If both posts are solid in the ground, the two braces should keep the gate from making the post it's attached to lean in towards the center of the walkway.
Good luck. Let us know how it turns out for you.
