We have a rescue Basset Hound and a rescue Cattle Dog mix. Both are nice dogs but VERY different! Both are good with kids, but the CD is hyper and jumps and the BH is lazy and stubborn, lol.
Basset Hounds are generally laid back and good with kids- but what I would mainly suggest is do some research online and then contact your local Breed rescue groups for WHATEVER breed you are interested in. Seriously, there is one for pretty much EVERY type of dog in almost every state!
The reason I am saying this is because not only will they know ALL about the breed, its health issues, quirks, etc. but the dogs they are fostering before going to a 'forever home' are living with people and they REALLY get to know each individual dog. It is not like you just walk in and say 'we want a Basset Hound' and they let you pick one!
They will ask you tons of questions - but remember, that is only because they really want the adoption to be a success!! They want to match you up with a dog that is a really perfect fit with your whole family and lifestyle. Later, if any issue do come up, the rescue group will totally be there for support and advice. Many of them have 'reunion' gatherings that can be really fun to attend. (ours has a Basset Hound Waddle, lol)
We have had many animals from shelters, and of course you can find wonderful pets there. But if you have more specific needs, a breed rescue will allow you to still help a homeless animal, but you can have a much more specific idea of what you are getting!
Check out rescue groups in your area! They all have websites- call them up and just tell them exactly what you told us and they will tell you if their particular breed would suit you. If it would, you can start the process. We ended up with a terrific Basset Hound. When we first drove to his foster home to meet him, my son (who was 7 at the time) got down on the floor to play with him. There were NINE other bassets in the yard outside,lol, both the owner's own and some other fosters they were caring for!
My son and Eli rolled around and played- the poor dog had been found on a road and was about 25 pounds underweight :( but then he just gave a huge sigh and laid down with his head in my son's lap and my son looked up and said " He will be my best friend forever!' and he has been!