We started at the bottom of the staircase and put treats up on each level, about 3 levels to start with. Pup feels more secure going up than down, so generally starting at the bottom gives them more confidence, and starting out with just 3 stairs doesn't make it so scarily high for them when they turn around to come back down. Gradually increase the number of stairs and soon he'll be taking them like a champ!
As for going from room to room--offer a handheld treat from across the threshold and call to Puppy in your best soft persuasive voice (it often helps if you crouch down). Of course, praise lavishly and give them the treat when they venture over the threshold.
Patience is your best tool here--but puppies are pretty adaptive and I suspect once he settles in your biggest problem will be to keep track of him as he runs from room to room and up and down the stairs.
Generally, imo, it's better to let pups explore and move around on their own instead of learning to rely on you to pick them up all the time.