Apple ID Switch/Split Here is some background knowledge... 
My Apple ID is also used by my brother. This originated back when the iPhones came out and it was necessary to plug them into a computer to start in order to house the music, apps, etc, that existed on my laptop. However, I no longer utilize this laptop or any other ones. 

Since we share an Apple ID we receive phone calls, texts, and voicemails for the other person EVEN THOUGH we've both gone into our settings and opted to receive only our own via our respective phone numbers and emails. 

I've recently received my upgraded phone.  I feel like now is the perfect time to split into two Apple IDs. 

My question:  How do I do this without losing all of my information? I use the iCloud extra storage for $.99 per month so all of my data is backed up, but i can't view it due to my lack of laptop (no need for one- until now).  What I am most concerned about is I want to retrieve my data from my current Apple ID and place it on my new one, while also being careful to not transfer his shared device to my new phone, which would result in the same issue as before.  Can I do this using iCloud without plugging into a computer?

I hope this rambling makes sense..  Also, I really do like my brother, I'm just tired of having double the contacts and messages, and a constantly full voicemail box.. I think you get the point. 

Thanks in advance!
An operation may be needed to separate the 'siamese' twins ;-)
It is not going to be totally painless. 

Start by investigating Family Sharing - which will enable you to keep the apps/music/books etc you have purchased.

Family sharing also necessitates designating one of the accounts as 'payer', so you may still have to negotiate who pays who for what, going forwards.

However, any data within each app, including anything like game progress, web &amp; other history, contacts etc is going to end up belonging solely to one brother. The other is going to have to start from scratch.

Contacts can be downloaded as .vcf files from the iCloud web page.
Calendars can be shared between family members.
...but I think just about everything else is going to be starting over from scratch.