Hi, we have always had dogs, our first two were lurcher/whippet type rescue dogs and were great animals. One didn’t like kids, but we were told this prior to any rehoming and this wasn’t an issue at the time. As the right questions, any decent rescue centre should allow a trial period first. Get as much background info into the animal, ie. How they came to be in the centre. A vet check is a must although some places will offer this…
As well as local rescue centres try looking at specific breed rescue centres. Greyhounds, for instance make great pets, but as you have a cat this could be an issue.
From what you have described a puppy may be best as the dog will grow with your family…
Saying that, we got our Border Terrier from online, she was a breeders bitch who had become too old for her intended purpose…she now lounges all day on the sofa and in front of the fire… A typical LBD..(lazy bloody dog)… :-)