Since I don't have the time to commit to keeping real pets, Nintendogs+cats is the next best thing. Best of all, the dogs are puppies forever.
Nintendogs+cats is primarily a pet sim which is suitable for short burst plays of up to 30 minutes. It can get boring if you play more than that since there's not much to do -- you are restricted to teaching a dog 3 tricks a day, you can enter the same competition (there's 3 kinds - lure chasing, obedience and frisbee) twice a day and toys and other goodies in the stores unlock as you earn owner points.
If you do want to see what the new 3DS can do, this is the best game to pick up since it uses all the new features -- AR card for Obedience competitions, voice recognition, pedometer (take the dogs out on walks when you do with the system in sleep), streetpass, spotpass (update and add-ons through wifi), etc. No other game at the moment uses so many new features of the 3DS, so it's worth getting for that. And the puppies are adorable.