You should receive a confirmation on the screen when the activation is complete. Does the phone work to make/receive calls? If so, then it is activated and should show as your phone when you log in to MyVerizon.
You need to back up the contacts from the old phone so they are stored on Verizon's servers; then use Backup Assistant on the new phone to import them. If they are not backed up with Backup Assistant, you can reactivate the old phone, run a back up, then activate the new phone and restore the contacts from the backup.