I am hesitant to do my first cemetery as I am not sure how to proceed. Who do I ask permission from if it's a public cemetery or a Roman Catholic cemetery? Can I just go in and start snapping pictures?
If it is a public cemetery, you don't need to ask permission. Start snapping.
You probably don't need to ask permission for a Catholic one either, although I have had to have a columbarian unlocked.
I've taken many pictures in Church cemeteries and other than a few people who ask me what I'm doing, no one has ever said anything
Always check if there is a sign at the entrance or business office, and if they say NO PHOTOS be sure to ask for permission. What this usually means it that they don't want people using the cemetery for "photo shoots" and recreational imaging. There are some that just don't allow any photos from the cemetery to be published to protect the dignity/privacy of the families. I know of at least one Catholic Cemees that has BG records but they are hidden.
If you are stopped by a grounds-keeper or maintenance person, politely leave and then contact the office/business for permission.
Welcome to the BG community. Take a few practice pictures in a cemetery with a lot of modern stones, until you get used to your camera and how to take good photos in variable conditions. Make sure you know how to do the transcriptions from your own photos so you can check the quality as you learn. Do about 50-100 at a time for a few sessions before you spend all day/many miles. Good luck and thanks for being part of this big work.