What kind of phone is your new one?
There are a few ways I have found:
1) I have read that you can touch the back of the phones together after activating the bluetooth and finding the phones within each others settings.
Go to Menu, Settings, Wireless and Network, then turn on Bluetooth on both devices, make one device discoverable, search for it in the other the once they are paired you can go to contacts and share them via bluetooth.
2) Copy contacts to your SIM card, then put it into your new phone and copy the contacts there. I have never tried this, but Google it and I'm sure you will see a solution.
3) Upload/Sync all contacts to a gmail account, make sure new phone has the same gmail account, then sync to new phone.
Hope this helps
Verizon says to download the Cloud App and here is how I saw it explained:
-Access the Verizon Cloud app
-Select the menu key located in the upper right hand corner.
-Select the Contacts Icon.
-From the drop down menu select "Back up Now" to restore the contacts on your new phone.
Once completed check your phones contact list to see if they have transferred over.
EASIEST way is to use a gmail account.
On your old phone use the gmail account to backup your contacts/emails etc
On your new phone, just login to your gmail account on your phone and your contacts are synced.
Verizon Cloud is good, but it also costs you if you sync too much and run out of space.
Google, and Google Drive is the way to go, IMHO