When I turn on the water in the bathrooms, the faucets and pipes shake and rattle.

You have a loose washer in a valve somewhere. Probably in the stop valve. Tell us more to help locate.

Tell us what your water pressure is, should be about 45 psi

If you have older style faucets with screw handles and washers, the washers could be loose. Does it happen with hot and cold or just 1? Does it occur with a toilet flush? Could also be a damaged pressure regulator with a cracked diaphragm.