You've gotten some wonderful replies here. Gardens and chickens are a natural, but that's not to say that they can coexist unsupervised. That said, my chicken coop is in my garden area. Here's a pic from last season:
As you can see, my three girls have a small outdoor run area that keeps them separated from the veggies. The whole area is separated from the rest of the yard by a picket fence. When I let the chickens out to free range, the picket fence is (ideally anyway) shut to keep the chickens out of the veggies. They won't go over the fence unless one of the girls needs to lay an egg. With only a few of them, I keep a calendar and only let them out in the yard when I know they don't need to lay an egg. Since I work from home, this isn't as difficult as it may sound.
Yes, chickens will eat all kinds of things. They love my beet greens. We had an issue where one of the kids left the picket fence open and the chickens had a nice lunch. They didn't dig up the roots, so we were able to just let the greens grow back. Mine will also eat rose petals but not enough that the plant is completely barren of flowers.
I keep a compost system running. The first is a pile where all coop cleanings, poop scooping, veggie scraps (those that aren't fed to the chickens), coffee grounds, and weed-free garden trimmings are piled. That's stage one. Then I run a ComposTumbler for speedier composting. I move the Tumbler stuff into my gardens (or that of my grateful neighbors!) and then rotate the more recent pile into the Tumbler. I've never had a problem with the poop burning my plants. The key to a compost pile is a proper balance of green and brown stuff plus just enough moisture (water/green things) and air (turning the pile) to make the decomposition process work.
Living in So. Cal, all of my soil is awful clay. I make up for it by having raised beds and supplementing with trucked in dirt (initially before I had chickens) but now only through my composting. So far it's working great and I think that the chicken poop compost has made my garden better than any dirt that I purchased in the past.
Like others, in the off season, my chickens have free access to the veggie beds. Any grubs hanging out in the soil becomes a tasty snack. And, any grubs that are eaten won't be around as an adult in my garden the following season. I have definitely found that I have less bugs in my garden with the chickens in the yard.
Definitely plant a little extra. Your chickens will thank you for a few treats tossed their way!