This question comes up from time to time.
Most public shelters will adopt a dog out to anyone who fills out the forms and pays the fee. With private shelters and rescues, it varies, depending on the individual family's circumstances, and most importantly, the individual dog.
A rescue needs something on which to base a decision when choosing an adoptive family; veterinary references and history of veterinary care of previous pets tell a lot about how a family cares for their pets. Previous training experiences, grooming experience in the case of high-maintenance dogs like poodles & doodles, are helpful. A dog who has any behavioral issues would not likely be placed with first-time owners. A well-adjusted adult dog who is being rehomed for economic issues, say, might be. But to increase your chances, you need to know the answers to the questions that you will find on an adoption application.
How would you resolve a digging, barking or chewing problem? What food will you feed? What vet will you use? How do you feel about the use of a crate? Where will the dog sleep? How often will the dog be walked? Which family member will be the person primarily responsible for the dog's welfare? (There must be one, and it must not be a child of any age.) Who will care for the dog when you are on vacation? You will be asked these things, and much more. In order to answer these questions, and more importantly, in order to properly care for and live with a dog, you need to learn as much as you can before you get the dog, not afterwards.
Start interviewing groomers, vets, trainers, etc. Decide who you will use. Start investigating various training methods to deal with normal misbehavior. Start reading about healthy diets for large breed dogs. Learn about various kinds of collars, leashes, and harnesses. Be able to prove that you know how to provide a good home for a dog. (I can also tell you that most good rescue groups would turn you down cold if you say that you are planning on having the children "contribute" to the dog's care. I cannot tell you how many dogs have entered the shelter system because parents got a dog for the kids and thought the kids would help be responsible for it.)
I am not trying to discourage anyone from adopting, but as a would-be first time owner without experience or first-hand knowledge of what dog ownership entails, your chances of adopting successfully will be greatly increased by being able to demonstrate that you have done your homework. Remember, an adopted dog has already gone through the pain and confusion of losing one home; the next one absolutely has to be the last one. Be sure that it will be before you adopt.
Do a lot of research & planning. Good luck as you start this very rewarding journey!