I have reset the network settings. SMS is turned on. I can only text other iPhones. This started back when I had an iPhone 4 and now I'm on the 6 Plus. I've even done a complete reset and still get nothing.
This sounds like a setup issue with your Verizon account and not the phone. Contact Verizon support and tell them SMS messaging is not working on your account.
are you using iMessage?
Yes
Have you tried to text the android users using the stock app? Does that work?
Yes. I've tried it with and without iMessage and SMS. Only using the text app that was on the phone from Apple.
I can text anyone with an iPhone on any provider but nothing else.
I can only receive texts from other iPhones also.
Kind of thought that myself.
Have you tried to remove your iMessage account from the cloud and adding it back? Did you have an android device in between the apple devices?
Fix iMessage: iPhone switchers can’t receive SMS — how to fix | BGR
Even if you haven't switched, you may consider still trying this method and telling the android users to re-add your number also.