Not sure what your exact issue is but if you want to restore your contacts from your old phone you need to...
backup your contacts from your old phone. Either in vz backup, icloud, or itunes.
Turn off old phone.
turn on new phone and either
get the vz transfer/backup app, follow the instructions,
connect to itunes (recommended),
connect to icloud. Note that if you have music stored on icloud it may not work properly on an iphone5 (i have this problem).
I'm not a verizon or apple support person. So I would get the instructions from verizon and/or apple to be sure.