The first  and most important thing I can say is to TAKE IT SLOW!!

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Your new dog undoubtedly has acquired stress, baggage, fear, uncertainty, confusion, and/or trauma. Even if he/she  has had a wonderful life up until this point, the mere fact that he is in a totally new environment traumatic and stressful for him. If the dog has come from a shelter, there are just that many more issues he has to  deal with. Remember that dogs only know what they've had an opportunity to learn with regards to interaction with humans. Don't expect him to know ANYTHING you haven't had the patience and forethought  to teach him.
 
A dog  craves consistency and security. Remove these things, and he naturally is not going to act himself. Even if the new dog acts like everything is fine, he has really shut off part of himself until he's surer of his environment.  The WORST thing you can do is confuse him even more, resulting in behavior which ends up causing  him to return to rescue or the shelter, and compounding his problems even more for someone else to (successfully or not) deal with. Be fair to your new dog. Increase his chances for a successful adoption, rather than decrease them.

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You will want to immediately show your new dog off to all your friends and family. You will want to take him to the park, to classes, or on outings. You will want to let your children and/or other dog(s) play with him. You will want to play off leash Frisbee or fetch. PLEASE DON'T!! 

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All these things require the dog to take in and deal with more than he is capable of at this early stage. It is not fair to expect him to be able to do this. Let your new dog have some down time and acclimate to his new environment at his own pace. In other words, let  him clean his mind out of the old stag as his body cleanses itself of the old food. As he clears his mind, gets used to things in his new environment, and his body acclimates to a new nutrition level, he will be able to think more clearly, and be ready for all that he needs to learn in his new home. 

Let him learn the daily routines you follow. Let him shed his fears and confusion before asking anything new of him, or  expecting him to make decisions. New things include: getting to know your children and any other dogs or  cats you have, tricks you want to teach him, invisible fence, where in the house he's not allowed, what not  to chew, who not to bite, etc. Allow the new dog at least a couple of weeks to just get used to you, and the way you communicate. DON'T ASK ANYTHING OF HIM. Do him the courtesy of not allowing  (forcing) him make any decisions by himself. He doesn't have the knowledge at this point to make decisions, because everything is new to him.
 
What you should do during this initial period is simple. Say his name lovingly and pet him when he asks for it. This  will get him used to the way you say his name, and associate it with only good stuff. During this time,  a simple "ah, ah" (followed by a high pitched "good-boy!", when he quits) is all you should use to discourage unwanted behavior (remember, he will not have the opportunity to do much wrong, because you are preventing it by not allowing him to be unsupervised or make decisions for himself. In other words, don't give him the opportunity to do anything wrong. This way, right from the beginning, you are establishing good behavior.
 
The idea of this is to put no new stress on the dog. Let him get rid of what stress he most certainly already has, before adding any. Every learning situation introduces a certain amount of stress to the dog as he tries to figure  out what you want of him. I don't think a new dog should leave the house and fenced yard, except perhaps for  a walk (keep it short, on leash, and follow the same route) for the first 2 weeks at least.