The first time I went to get a dog, we went to a pet store. I would never do that again - for reasons that are another post. I would encourage anyone who is looking for a dog to check out their local shelters first. If you are in Atlanta, one I recommend is Atlanta Pet Rescue. They have all sorts of dogs and cats, a good variety of breeds and ages. If you are looking for a particular breed, all you have to do is wait and be patient.
Another option is finding a pet through Craigslist. This is where I found my dog, Zoey. Craigslist is more individuals who are giving away their pets due to various reasons - not enough time to spend with the pet, health issues, moving, etc. One entry I read was about an owner who was about to enter drug treatment and wanted to make sure his pet had a good home. Craigslist can be great also if you want an animal other than a dog or cat. Again, you have to be patient but it's worth it when the one you want comes along! You can also check it for pet supplies, like cages.
A lot of people want puppies, but there are good reasons to consider getting an older dog, especially if you've never owned a dog before. Older dogs are less hyper, you can better tell their personalities, and you don't have to deal with potty-training issues as much. Of course, the advantage of puppies is that you can shape them to a certain extent through exposure to various things at a young age.
As for cats, I personally love kittens and probably wouldn't go for an older cat. However, older cats need homes too and they have their advantages. Like dogs, you can see what their personality will be like and they'll be less wild and destructive. While kittens are really cute, they can also be really annoying when they want to play Chase the Toes in the middle of the night. :-) (Or, if they're like anything like my cats, they'll want to either hold races around your bed or they'll want to purr loudly and knead their sharp claws on your stomach. In the middle of the night, of course.)
So, next time you are in the market for an animal be sure to check out your local shelter!