Highly polished stones are shiny, but watch out for reflections that obscure the writing. Move around the stone, high, low, left, right, and find the angle(s) with the least glare.
Hey, is that ME in there?
Be aware while you are looking for an angle with the least glare that angles that reflect your image in a mirror will also reflect your image in any smooth surface.
Look carefully at this photograph. Because this was taken from a straight-on view, you can see the photographer kneeling to take the photograph.
Yes, this does happen. Sometimes, when the stone is very smooth and the light is just right, you can tell what camera the photographer was wearing! So although you do not need to be at the height of fashion to go hunting down headstones, the clothing that you wear should be appropriate. Think flattering, not frightening!
On the other hand, it could be quite artistic to include your reflection with your image on grandpa’s headstone, making a family photo of sorts. I do not want to discourage your creativity, but I do want to raise your awareness about reflections.