I am glad to see that I'm not the only person having these problems. I have spoken with Apple Support twice and Verizon Support twice. If you're interested, please read on:
I noticed right after I got the phone that I had a random 64 MB download on cellular data at 10:30AM. Strange, as that is right around the time I watched a YouTube video while connected to my home WiFi to see the beautiful new screen in action. I didn't think much of it and used the phone as normal on my WiFi network at home and work. I specifically chose not to use it while connected to cellular to try and minimize my usage. A few days later and, perhaps, 5 websites and 20 e-mails on cellular and I've already accrued over 200 MB of cellular data usage. While this may not sound like much, when you consider that in the previous month, while actively using plenty of cellular data I used 350MB, it seems kinda high for 5 days. I went back to verizon's website and checked my data usage, and sure enough there was another 70MB at 8:30PM on Friday. Shocking as that was right around the time I watched my 2nd youtube video while connected to my home WiFi. I also monitored my cellular data usage through the Settings -> General -> Usage -> Cellular Usage and found throughout the day on Friday and Saturday (I was home all day both days and connected to WiFi) I was constantly using cellular data.
At this point, I am starting to get concerned about what is happening here. There is no way I have used up that much data checking a few webpages and e-mails. I went into a verizon store to speak with a rep and was essentially told that A) When the phone is connected to WiFi, it does not use cellular data (Yeah, I get that is what is supposed to happen, thanks) and B) LTE uses more data to do the same thing on 3G (Not even remotely true, 400KB through LTE is still 400KB only transmitted at a faster speed). I could tell I was going to get no where with this, so I left disappointed. I called Apple Support and they confirmed for me that the phone was properly connected to a WiFi network. She suggested that I do a restore on the phone and see if it was still happening. I did and it was.
I have now been turning off cellular data in my settings whenever I know I will be using the phone for something on WiFi, but this shouldn't need to be done.
This brings us to today. I call Apple Support again to see if this is being recorded as a persistent problem and will be corrected. I get bumped up to a senior advisor and she walks me through my settings and verifies that everything should be fine. She puts me in touch with a top-tier verizon support specialist so that I can document that I'm having these issues. She tells me that the times listed on the verizon website are not always accurate and that it may just be the day's data usage catching up to me. Strange as there are days that I never left a WiFi network and still had plently of cellular data usage. After confirming that I know what I'm talking about, she agrees to monitor it and contact me in a week to look over everything. In an effort to track it, I have to dial #832 everytime I am about to connect to or disconnect from my home and work WiFi. While this will be an inconvenience, perhaps it will actually show that there is a mismatch in what the phone should be doing and what it actually is doing.
If you read through all that, you are a trooper. Hope my experience at least informs if not helps you with your own.
I will report back next Saturday when I have my followup call with Verizon.
P.S. This was more of a cautionary situation. I don't want to find myself in the situation of unknowingly going over my data when it isn't my fault.