If you would like to adopt a dog that isn't a puppy, I would suggest looking at www.petfinder.org there are a lot of different rescue organizations with tons and tons of great dogs looking for homes. Some of the rescue organizations don't allow families with children to adopt the dogs but there are a lot that do. You can search right in whatever your criteria ends up being. Plus if the dog doesn't end up being a perfect fit for your family, they usually will often re-home the pet.
Puppies are great if you are able to take the time and have the energy to train them. I highly recommend taking a puppy to a puppy manners class and then one or two obedience classes after the puppy class. It will really help you to make a great addition to your family.
As far as breeds are concerned, if you don't mind the extra hair and shedding, Golden Retrievers are the way to go... they are the absolute sweetest most forgiving dogs out there and they absolutely love to be a part of a family. They are very much a happy go lucky dog and love to play, fetch is their favorite which is great with the kids. We have a Golden and got her when my son was only 3, he was afraid of dogs at the time, well our Golden definately changed that, they were very quickly the bestest buds and he can do just about anything to her and she doesn't mind at all, she just loves all the attention. Also, any Lab type dog or Lab mix are great too. You can also search online for a breed you like that is good with children. Here is a website that I really liked that rates dogs with children (I copied the link to that area but the whole website has tons of great info)... http://www.dogbreedinfo.com/children.htm
I would steer clear of any smaller type dog as they tend to get nervous around kids and tend to be nippy. I have had dogs my whole life and have had quite a variety of differnt breeds as well, I have trained quite a few up to the competition obedience level. Hope that helps!