Obedience training usually refers to the training of a dog and the term is most commonly used in that context. Obedience training ranges from very basic training, such as teaching the dog to reliably respond to basic commands such as "sit," "down," "come," and "stay," to high level competition within clubs such as the American Kennel Club, United Kennel Club and the Canadian Kennel Club, where additional commands, accuracy and performance are scored and judged.
Once your dog is ready to start agility training, your best bet is to find a class or group in your area. The United States Dog Agility Association (USDAA) offers a directory for agility groups in each state, and many dog trainers offer classes as well. In the classes, you will be able to introduce your dog to the obstacles without the expense of buying or building them yourself.
Dogs that demonstrate the previously mentioned basic skills, as well as walking reasonably well on a leash and a few other minor tasks, can be tested for and earn the American Kennel Club's (AKC) Canine Good Citizen certification. While not a competitive obedience title, a CGC certification demonstrates that the dog is sociable, well behaved, and reliable in public settings. Some insurance companies will waive breed restrictions on dogs with CGCs, and many states have passed resolutions supporting and encouraging CGC certification as a yardstick for canine manners and responsible dog ownership.
Remember that training is an ongoing process. You will never be completely finished. It is important to keep working on obedience training throughout the life of your dog. People who learn a language at a young age but stop speaking that language may forget much of it as they grow older. The same goes for your dog: use it or lose it. Running through even the most basic tricks and commands will help them stay fresh in your dog's mind. Plus, it's a great way to spend time with your dog.