It's metamorphic and might be a variety of Serpentine, but testing and microscopic examination by a geologist is the only way to be sure. If you have a college nearby that offers courses in geology you might be able to do a walk in and get a definitive answer.
Not metamorphic. Not a serpentine or travertine. Looks like plain old sandstone to me.
The rings are what's called Liesegang rings. There's a whole explanation of what is thought to cause them that'll even bore geologists. But, in a nutshell, it's a secondary sedimentation caused by a chemical weathering………….usually the oxygen in water leaching ferrous (iron) compounds and re-crystallizing.