Adopting a dog from a shelter is a great option. Animal shelters have a variety of dogs available of different sizes and breeds. A good shelter will provide behavioral testing and initial vet care also, so that your new pet has a head start on their health when they come to your new home. Many shelters list their available dogs at Petfinder.
Shelter workers usually don't have a particular breed of dog they're interested in promoting ahead of others, and they hate returns more than anything, since that shows that a match between the dog and its owner wasn't successful. A good shelter will always be ready to take back an animal if it doesn't work out. Because of their aversion to returns and independence of recommending any particular breed, they can often recommend a breed or type of dog that you should consider based on your lifestyle and preferences. I also like the dog breed selector by Animal Planet.
If you do already have a specific breed in mind, then contact a breed rescue group. There are rescues for every kind of breed out there, and most are findable by Google.
Some people find that shelter dogs show a lot more gratitude for their home, too.
Training an older dog can be a concern. I've personally had no trouble, and so long as the dog's motivation can be found any dog is trainable. I've trained 10 year old Chihuahuas to dance, a 3 year old Coonhound to sit at attention, and a 2 year old beagle to sneeze. All of these were dogs from the Humane Society of Indianapolis.
Shelter dogs often come with a pet health insurance policy at adoption, which is a wise way to limit your risk in adopting a pet. I do know of one person who rescued an abandoned puppy and spent a lot of money on vet care. Going through a shelter to adopt makes financial sense, again because they have screened the dog for health issues and provided initial vet care. They sometimes miss things, that's true, but many shelters provide a pet care insurance policy free for the first 30 days to cover anything they have missed which your vet finds. We use it for our dogs, and if you mention us as a referral we'd be happy to donate the $25 referral bonus they pay us to an animal shelter of your choice.
For some, a shelter can be an overwhelming place. If that is true for you, contact your shelter and see if they have any foster dogs available. Often a foster parent can work with you to adopt the dog from a home, and you still get the benefits of having the dog's initial screening and vet care completed. Plus, to get along in a foster home most dogs will learn a few house-manners and even some training.
I've included some photos of dogs we've fostered to show what great variety of dogs are available. For more info about our foster dog's experiences, check out the former fosters blog.