If you want to get a phone that is not locked into service from a particular carrier, you may want to buy your phone directly from Apple themselves, and avoid any of the payment options that involve carrier financing.
Look for words like "And your new phone will be unlocked" – words which appear or disappear depending on how you indicate that you are going to pay for the phone.
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Once you select, and have, your new phone, these articles may be helpful:
I believe that it is also possible to do a transfer by backing up your old iPhone, restoring the backup to the new one, and synchronizing the new one with whatever you were synchronizing the old one with before (e.g., iCloud Photos, e-mail accounts, etc.). But if you have both phones handy, Quick Start might be easier.
I do not believe that any of these methods transfer purchased music from sources other than the iTunes Store (like your own CDs, or other online music stores). If you had used manual synchronization to put purchased music onto your old iPhone from a Mac or PC, you could do the same thing to load music from the Mac or PC onto the new one.