I upgraded to an iPhone 5s for Verizon and my data is going crazy

Sorry this is long, please read. I used to have an iPhone 4 on a shared unlimited data with Verizon and 2 family members on it too. And it was cool because I only used like 1.2 ish GB of data a month.
So when I upgraded yesterday to an iPhone 5s, I chose the 2 GB plan because they no longer provide unlimited.
Anyway, it's been about 20 hours since I opened and activated my phone and there's already 160 MB of data used! At this rate I will be doing like 6 GBs a month! Did upgrading to LTE mess with it? Do phones differ with data? Because I'm using it about the same amount as before! (Yes I did download like 15 apps because it is a new phone but they were all over Wi-Fi--which I thought was free)

If you are downloading apps and things use Wi-Fi but, go to settings and turn off cellular data. You should be fine.

You can also turn off background refresh for any apps you don't need to be constantly refreshing. Settings > General > Background App Refresh

LTE does not, in and of itself, use more data than 3G. However, many people due tend to end up using more data simply because LTE is so-o-o much faster than 3G that they can do more in a shorter period of time. You may think you're doing the same thing as before, but I guarantee you're not. You can actually look in the settings on the 5S and see a list of apps that are using the data. I believe it's Settings>Usage (not 100% on that but it's in Settings *somewhere*). If I remember correctly (I don't have an iPhone in my hand at the moment), there's also a setting in IOS7 (latest iPhone software) that basically says if the phone detects a weak Wi-Fi signal, it will automatically switch to LTE instead so you may have been on LTE when you thought you were on Wi-Fi. Thus 'sassy's' previous suggestion was also a good one!

Hope this helps!