Amador City has a legacy of antiquing. You and your dog can walk the outdoor craft mall that has shops lined up and waiting for you. Some of them will allow you to bring your dog in, although some are breed restrictive. Please bring your leash and plenty of water for you and your dog. There isn't much shade, but there are plenty of seating areas.
If you are looking for something to fulfill your sense of adventure, then you could try mining for gold! There are many mining areas that are family-friendly. You can walk the trails with your dog, but they aren't allowed in panning areas or allowed to splash in the water. Your children can't play in the water, either.
You can find more conventional dog-friendly spaces just a short drive outside Amador City.