How do I get my contacts to come over to a new phone?

I moved from att to Verizon and went from a 7 to an 8 iphone. I backed up the phone, logged in and transferred what I thought was everything. However none of my messages or contacts came over......

How do I get them to come over to the new phone since I must turn the 7 back in to att and would like to keep all my messages?