Can I transfer content from an existing iPhone to a new iPhone which is using a new (different) Apple ID? I am trying to cut myself off from the family shared AppleID. I've just bought myself a new Macbook and have set up a new Apple ID. I have also just got a iPhone 5 to replace my dying 4.

My question is, can I back up the contents of my iPhone 4 and then bring that content onto the iPhone 5 with the new apple ID. I understand that I'll lose the apps that would need to be purchased again, I guess I'm more concerned about the contacts, notes, and photos.

Any advice would be much appreciated!
I'm a little hesitant to post this because I can't try it right now and there is, as always, a chance that it won't work. But, because you are willing to repurchase apps, the transfer should be fairly achievable. 

I would first restore from a backup of your old iPhone to your new iPhone. If you have already setup the new iPhone then you would need to wipe it and start over which would erase any data that you have created since getting the new phone. Through this process, you will sign in with your old Apple ID. After everything is restored, I would delete the iCloud account off your phone. It should ask you whether you want to keep notes, contacts, calendar, etc. choose to keep all of those items. Sign into your new Apple ID and create a backup of your data. 

Again, as I'm sure you are aware, you will still have to work through your apps and repurchase or download all of them again. Apps like Evernote are easy because your data is in your account and you would just have to sign back in. Other apps like games are notorious for losing data.