Quickest, easiest way is poison. In that setting, it would need to go into "tamper resistant bait stations" to keep it away from kids, pets, and (most) non-target wildlife. With a decent station and a little common sense, your dog will be perfectly safe.
Believe it or not I have found time and again the best traps for them are the same live (cage) traps used for squirrels and chipmunks. Any non-target animals can be released. When a rat is caught, pop it in the head with the pellet gun through the mesh. The oil type sunflower seeds and/or dry dog food are best baits.
If you try to hunt them, they will figure that out very quickly. They really are very smart animals.
Good luck!