Verizon offers a program called Backup Assistant that allows you to move your contacts from your old phone to your new phone. There are also separate cloud-based services that allow you to back up your old phone and then import contacts to your new phone.
If you are using an iPhone, you can back up your contacts to iCloud. To do this, go to your settings and click on the iCloud option. Once there, turn on the contacts option. This automatically backs up your contacts to iCloud. Next, log into iCloud with your new phone, and iCloud automatically downloads your contacts to your new phone.
If you are using an Android phone, you can use the Verizon Cloud option. If Verizon Cloud is not already on your device, you need to download it from the Google Play store. After downloading, open the application and select all the options you wish to back up. Once this is complete, you can activate your new phone and log into the same Verizon Cloud account. This will allow you to access and download all of your contacts to your new phone.
Additionally, if you would like to avoid cloud-based services, you can back up your phone to your computer. You simply need to plug in your phone using a USB cord and select the pieces you would like to back up. When you've done that, you can plug in your new phone and import all your contacts.