Moving “Palmettos” & Gardenias

I live in Kenner, LA which is right outside of New Orleans. I have two plants (see behind the cut) in my front flower beds that are called palmettos by people here. I am not sure if that is their correct name. I would like to move them from my front flower beds and replant them in my backyard. I am wondering if anyone can tell me how difficult this process might be. How deep will the roots be and how big are the root balls? Is this something that I could accomplish with my husband or will we need additional assistance?

Also, I have a gardenia (shown in the left flower bed photo) that I would like to move. It currently has buds forming. Will it harm the plant to move it right now?

Thanks in advance for your help!