Answered question
I have an iPhone 5.I don't have a data plan,but I do have unlimited phone and text service. How do I text on my iPhone when I am not connected to wifi?
Texting (SMS) as well as multimedia messaging (MMS) are separate from data (which is access to the internet.
Your problem is you're using an iPhone which defaults to using the apple iMessage service or whatever its called, which always uses the Internet for message sending. You will need to contact apple support to figure out how to make it send real SMS messages.
Other answers
You don't need data plan to text.
In the settings app > messages > Send as SMS.
Toggle that on and it’ll send through text message when not on wifi.