Be Aware of Body Postures: One body posture that indicates things are going well is a "play-bow." One dog will crouch with her front legs on the ground and her hind end in the air. This is an invitation to play, and a posture that usually elicits friendly behavior from the other dog. Watch carefully for body postures that indicate an aggressive response, including hair standing up on one dog's back, teeth-baring, deep growls, a stiff-legged gait, or a prolonged stare. If you see such postures, interrupt the interaction immediately by calmly getting each dog interested in something else.
Taking the Dogs Home: When the dogs seem to be tolerating each other's presence without fearful or aggressive responses, and the investigative greeting behaviors have tapered off, you can take them home. If you have more than one resident dog in your household, it may be best to introduce the resident dogs to the new dog one at a time. Two or more resident dogs may have a tendency to "gang up" on the newcomer.
Dogs have definite heirarchies, and it is important to support the dominant dog in your household, even if that turns out to be the newcomer. This may mean, for example, allowing the dominant dog to claim a favored sleeping spot as his or to have access to a desirable toy. Trying to impose your preference for which dog should be dominant can confuse the dogs and create further problems.
Puppies: If you're bringing a puppy home, the puppy may not recognize subtle body postures from the resident adult dog signaling that they've had enough. Well-socialized adult dogs with good temperaments may set limits with puppies with a warning growl or snarl. These behaviors are normal and should be allowed. Adult dogs who aren't well-socialized, or who have a history of fighting with other dogs, may attempt to set limits with more aggressive behaviors, such as biting, which could harm the puppy. For this reason, a puppy shouldn't be left alone with an adult dog until you're confident the puppy isn't in any danger. Be sure to give the adult dog some quiet time away from the puppy, and some extra individual attention as well.