I would just use itunes as its simple and fast. Just follow these steps.
1) Connect your old iphone to itunes and click backup. This backs up EVERYTHING.
2) Wipe your old iphone (erase content and settings), always do this to wipe it clean, this also removes your icloud password so if selling it someone else can set it up as their own. Do this even if you are trading it in.
3) Get your new iphone, trade in your old for credit if you want. When you setup your new iphone, you will connect to itunes on the computer, and it will then walk you through the steps. Choose yes to load from backup, when you load from backup it will copy over everything so its just like before.