How important is it for my puppy to play with other dogs? Who should my puppy play with? What should that play look like?
These are all important questions to ask when socializing your new puppy. Ideally, get your puppy into a GOOD puppy class before the age of 10-12 weeks so that you can have some playtime with other puppies in a well-supervised setting. Look for an instructor that will actively supervise the puppy playtime and stop play that is overly rough or frightening for any of the puppies. Ask questions about why the instructor is or isn't stopping play so that you can learn what is acceptable play and what is not.
In general, you want your puppy to play with other puppies very close in age (even a few weeks older than your puppy may make a big difference at this age!) or with good adult dogs that are calm enough to not overwhelm the puppy. For example, if your puppy is 9 weeks old, she should not play with a 16 week old puppy or an adolescent 2 year old Labrador!
As for what the play should look like, it should be fairly even. Both puppies or dogs should spend some time on top of the other rather than one puppy always being pinned to the ground. If one dog looks fearful (cowering, hiding, etc.), give it a protected area to stay during initial play sessions. An exercise pen or baby gate works well for this. If you're uncertain about the quality of play, pull the two puppies apart and hold them apart for a few seconds. Then, let them go and see what happens - if they both run back eagerly, the play is probably okay. If one seems less eager to return to the play, or even relieved at having the play stop, it was too rough!
Last, stay out of the dog parks at this age!!!!!!! Not only is your puppy at risk in a dog park because it is not fully vaccinated, you have no idea what play style and manners the dogs at the park will have. Your dog may be exposed to bullies and dogs that will teach it very poor play rather than teaching nice manners. Stick to puppy classes or play dates with pre-screened dogs in your own backyard until your puppy is at least 6 months old.
Cara, Buddy's Chance, Austin Dog Training, Dog Behavior, and Dog Daycare