I'm hoping you all can help me

I just got my baby yorkie this past weekend and he's 9 weeks old. Obviously he's too young to go to the groomers as he has not had all of his vaccinations yet and his vet doesn't want him to be in that kind of environment with other animals until then. My baby Oliver's coat is really getting matted and I brush him everyday but it still just doesn't look soft and clean. I'm going to give him a bath tonight with this new puppy shampoo/conditioner that his vet recommended but will that make him look soft and fluffy like a puppy should look like?? I'm also afraid to trim his nails myself because I don't want to hurt him. What did you all do for your little one when he was this young?? Thanks!!