switching to a new apple id and transfering it on to a new iphone

Hi, I have a shared apple id with my mom on my old phone. I also have my own (new) apple id i created for my mac. I want to transfer all my data to the new phone but i want to be signed in with my own apple id (the one I use for my mac). Is there any way to do it? I was thinking of backing up on iCloud but i don't know if I should buy extra iCloud storage through my new apple id or through my moms apple id (the one I'm using on my phone with all my data).

Thanks