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
. . . and growing and growing and growing. . .

. . . and growing and growing and growing. . .

Plant ID

Plant ID

Could it be honeysuckle?

Could it be honeysuckle?

on elephant ears and rubber plants

on elephant ears and rubber plants

Deer and the Garden

Deer and the Garden

My small room ~ tour ~ (img heavy)

My small room ~ tour ~ (img heavy)