Define "everything" in the context you're using it. As for your messages, do the following:
1. Reboot your iPhone 5C.
2. Make sure it is connected to the Internet, either via Wi-Fi or cellular data.
3. Make sure that iMessage, SMS & MMS are enabled in the Message settings.
4. Check to see if your cellular provider is blocking your text messaging capabilities.
5. Try resetting the "network settings" on the device.
Have you done EVERYTHING that @Just_Me_D recommended, including contacting your carrier? If you have and you're still having issues you may need to restore your iPhone. I recommend using iTunes to restore. http://m.imore.com/restore-iphone-ipad-itunes
Yes, you would have to contact Verizon if they are your carrier. As far as restoring your iPhone, I left you a link with instructions in my previous post.
I couldn't tell you that for sure but if you're able to make phone calls and access the internet I don't think it would be a SIM card issue. Have you spoken with Verizon yet?