What Dooin said. Staffs can be a bit mad, you don't always find calm ones at age 4 or so, but when you do they're a joy. Greys are notoriously laid back but I have known some to suffer seperation anxiety (as can any dog, even if you have him from a pup) so pick carefully with this in mind as you don't want to alienate your neighbours.
But, mainly, Dooin's number one choice. The calmest, most laid back and family friendly dog I've ever owned (bear in mind I've fostered lots too) was a HUGE great long haired German Shepherd. He came to me when I'd lost my little Collie Cross girl to cancer and was looking for a similar sized (ie Cocker sized) bitch. Didn't find one of those but met Shane the GSD. A match made in heaven and I KNOW that no matter how long I live and how many other dogs cross my threshold there will never be another who will be quite as perfect.
If you go to a reputable rescue they won't say "have a look and take your pick". They'll ask you everything but the colour of your knickers and then suggest one, two, maybe three dogs who they think will suit you and your family.
And if you do go through the rescue route, as I hope you will, remember please to only go to one which will question you at length, homecheck you, insist on meeting all the household and any other pets, neuter, vaccinate, assess their dogs thoroughly and take him back at any stage in his life if heaven forbid you couldn't keep him.
Remember please too that an ex stray is no more of a "risk" to your family than a rescue dog who has landed up there via a family home, provided that the rescue is reputable. Regardless of where he has come from they should thoroughly assess him. A good rescue won't just offer you a dog with "a history" behind him because they know he lived with DC and cats, they will have taken in all that the previous owner has said and then assessed him anyway, just as they would a former stray because, sadly, we know that owners will sometimes lie.
Lastly, ensure that you are adopting from a RESCUE and not a pound. Ask outright... "Do you have a contract with the council to take in strays?" because some pounds will go under the name of "rescue" or "dogs home" and try to make out that they ARE rescue. The problem with this is that they are only paid by the council to keep dogs for 7 days, after that they have to pay for the dogs keep themselves if he is to stay longer and as a profit making concern that isn't in their interests. Hence many pounds, if they don't kill the dog or send him to rescue, will be anxious to get him out, which is fine if they carry out good rescue practice and neuter/assess/homecheck etc... the problem is many don't and will just say, there he is, you can take him today, he seems fine to us, that'll be £100 please. AVOID these like the plague and let us rescuers help those dogs by taking them in and assessing and rehabilitating if need be, don't YOU take that risk, please.
An example of a pound which will rehome responsibly (though as a rescuer I don't like them because they put to sleep or reject many dogs) is Wood Green Animal Shelters. They are a pound and they take in the councils strays BUT they do all the necessary assessment and homechecks before offering dogs for adoption so they're one of the few pounds I would say are okay to approach.