First try your local animal shelter. When stray dogs are captured, or pets are lost, they usually end up at those shelters, and if the dog is 5 years old, they will be euthanized if not adopted by a certain time. Ask which dogs are about to be euthanized. I once found an adorable little black terrier at an animal shelter in Quincy. He was on his last day of life.
You can also check out animal adoption fairs. They are sometimes sponsored by local pet stores, and if you go to a centralized location, you can see all the dogs available for adoption. They are usually advertised on local news media.
Before you adopt a dog, you should always try to research the breed, if the dog is a purebred, to decide which sort of dog you will get along with. My former sister-in-law adopted a Jack Russell terrier which was so hyperactive that it almost never stopped jumping straight up in the air. He didn't live with her for long.
When my rescued cat passed away, I found an ad on Craigslist for a veterinarian rescue league in the far western part of my state, where they were giving out cats and kittens THEY had rescued from a kill shelter for $200, which was the cost of spaying and a microchip. I drove there, 3 hours and back, and found the perfect cat for me.
The ASPCA might also have a chapter in one of the large cities in the state where you live.
And don't forget to check the bulletin board at your veterinarian's. They often allow people to advertise their dog or cat who needs a home on their public bulletin board. Or they might allow YOU to put up a 3 x 5 card advertising your search.