Moving “Palmettos” & Gardenias

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!
Left Flower Bed Left Flower Bed
Right Flower Bed Right Flower Bed
Kitchen Details

Kitchen Details

Furniture Arrangement

Furniture Arrangement

Slugs? Or… something with teeth? Help me out here.

Slugs? Or… something with teeth? Help me out here.

Darlingtonia!

Darlingtonia!

ID a bug

ID a bug

Azalea

Azalea