That's a big leap in phones! My advice would be to take the photos, videos, and music and save them to an SD card, and from there to your computer. The contacts should be backed up using Verizon's backup assistant on the phone.
Once you get the new iPhone, see if a corporate Verizon store is able to transfer anything for you. Contacts may be all that they will be able to do, but if they can transfer your contacts, that's a huge help.
If the store is unable to transfer anything, you can use Backup Assistant or the Verizon cloud to get your contact list. Then use your computer and a USB cable to move the photos & videos. You might be able to use iTunes to get the music onto the iPhone, but it may not be in the correct format for iTunes - someone else who has moved from a basic to an iPhone may have a better way to get it all onto your new phone.
When I went from an env3 to a smart phone I transferred contacts and photos,with bluetooth and it worked nice. Mary