When they are first getting used to you they are shy and nervous, this is the 'honeymoon' period when they seem to be on their best behavior, then when they are comfy their true terrier comes out!
Don't worry, your puppy is just being a puppy, he is going to push your buttons and try and see where the rules lie. Get him lots of chew toys, he will be teething for awhile. Make a loud 'ouch' when he bites hands or fingers and stop playing for a moment, redirect with a toy and praise him for bitting that. As for the barking there are lots of things to try, when mine was younger it was mostly just an outlet for energy, I started exercising her more and it helped. Basic obedience class also worked wonders, as soon as possible sign him up for puppy kindergarten. Good luck!