Congrats on upgrading your device! I'm sure you will love all the features that the Iphone 4 has to offer. May I ask have you downloaded the VZ Contact Transfer app onto your Iphone 4? If not, then I would recommend that you do so. This is the app that equivalent to backup assistant.
Once you have downloaded the app, launch the app to begin restoring your contacts. Press Next at the top right hand side of the screen to accept the terms & conditions. On the next screen you want to enter your mobile number that you want the contacts restored from & press OK. Once you have done so, you will receive a text with your access code. You want to enter the access code, usually 8-digits long, when prompted to restore your contacts to your Iphone 4. The process takes about 2-10 minutes, depending on how many contacts you have.
Enjoy your new device.