Let me say first that Therapy Dog work is every bit as much about the handler as it is about the dog. If your dog has what it takes to be a Therapy Dog - and it takes a special dog to truly be good at and enjoy this type of work - you also need to have what it takes to be a good Therapy Dog handler, because it's up to the handler to introduce his (or her) dog, keep up conversation, and interact with the people you are visiting as well as the staff of where you will be visiting. You will meet all manner of people. Some don't like dogs and don't want visits. Others will want to tell you their life's story. Some staff won't see the point in Therapy Dogs.
There are three national organizations that test and register Therapy Dogs - Therapy Dogs International (TDI), Therapy Dogs Inc. (TDInc.) and the Delta Society. Each has their own requirements, and requirements do vary by organization. Then there are, of course, many smaller, local Therapy Dog organizations which will have their own rules and testing.
Both of my dogs and I belong to Therapy Dogs International (TDI).
In order to test with TDI and become a registered Therapy Dog team through them, your dog must be at least one year of age. You have to find a test - you can get test dates and the contact information for TDI evaluators through their website at Therapy Dogs
During the test, you will have to demonstrate that your dog has basic obedience skills and that you are in control of your dog. Many of the test points are the same as the AKC Canine Good Citizen test, so if you were to find a local CGC test and you pass that with your dogs, they will probably do well during the TDI test, too.
Testing (and visiting) needs to be done on a flat collar or a regular harness. TDI does not allow any head halters, front-clip harnesses, or training collars during testing or during visiting. I do believe they allow all-cloth martingale collars (not the nylon and chain kind) as well, but I am not 100% certain as I don't use one of those and have never specifically asked about it.
The way you test depends on the tester. Some test each team individually, like they do for most of the CGC tests, and some test all teams together. Personally, I prefer the second version, as you have to walk in a crowd, pass other dogs, etc. as opposed to simply meeting a friendly/neutral dog or passing through a few people.
During the test, you have to demonstrate sit, down, stay, come, and walking nicely on a loose leash. You have to also demonstrate that your dog will sit (or stand) politely for petting, knows a "leave it" command to refuse food, and is willing to go up and interact with people to be petted - even if people are using crutches, wheelchairs, or other "weird" medical equipment. Your dog also has to stay with someone for three minutes while you are out of sight without barking or whining.
Once you pass the test, you are given a form to fill in and send back to TDI along with your dog's medical checkup. Your vet has to fill in the medical form, and it's pretty self-explanatory.
After sending everything in to TDI, you will get a packet in the mail containing your member's guide, a DVD, your dog's ID card, ID tag, and TDI bandana. There will also be a couple of different forms in there, including a copy of the insurance policy you and your dog will be covered under while volunteering as as TDI team and some brochures to hand out (one will be about the TDI test, one will be about Therapy visits and one about the Tail Waggin' Tutors program).
You will also get a list of facilities in your area that are already receiving or have requested visits by TDI dog-handler teams, and a list of chapters if there are any in your area. Once you have received that, you are able to begin visits. If there are no facilities listed in your area, you can approach them yourself and ask if they would like to receive visits.