What are the basics of dog training? Rescue dog with training already? please read whole question?

I want to get a dog from the Humane Society and will opt towards an easily trainable dog and/or one who is already trained. If he is already trained, what will I personally have to do to so he responds to me? Also, what exactly are the things I would train him? Basically, if he listens to me and can understand the commands, he will be able to understand most of what I want him to do, right?