Sorry to hear about your dog, shit always sucks. I think it really depends on the person. I've known people who have waited around a year or more and there are some that have gone a few weeks then get another. My personal opinion would be wait at least 3-4 months, especially since it is your first pet. Just remember the good times you and your family had with the dog, your house will feel empty, I even got myself walking around where the dog laid in my bedroom as if she was still there out of habit.