The problem with cloud services like Verizon Cloud is that it might be hard to use that stored data once you leave Verizon. That's why it's a much better idea to use Google Drive and Google Photos to backup/sync your data. You may want to use Google Photos to backup all of your photos right now:
https://support.google.com/photos/answer/6193313?co=GENIE.Platform=Android&hl=en. Once they're backed up to your Google Photos in the cloud, you can access them from any device that's logged into your Google account, and you can download any of those photos locally if you need the actual files.
Are your contacts also backed up to Verizon Cloud? Do you know if the original contacts are also saved to your Google account? Go to contacts.google.com on your computer browser and see if the contacts are there. If so, then they'll sync automatically to the new phone once you log into it with the same Google account. It's trickier if those contacts were saved only to the Verizon account or to the local Phone account, because those accounts don't sync to Google. The Verizon account syncs to the Verizon Cloud, and as I mentioned, you might have problems accessing it once you leave Verizon. The local Phone account doesn't sync anywhere. For those accounts, you'd have to try exporting the contacts to a .vcf file, then importing that file into your Google Contacts, as discussed here:
http://forums.androidcentral.com/ge...-android-updated-2016-nougat.html#post3707512
The OnePlus 3T has a USB-C port, so if the phone doesn't come with a USB-C to USB-A cable (USB-A is the older, more common port), then you can buy one easily to connect the phone to your computer. This will let you transfer files from the computer to the new phone via USB connection.