TextMe Up is a free service that works over your existing internet data plan (the same you use for email and web browsing).
You still have to check your phone operator contract for your data plan details, but if you have unlimited internet access from your smartphone, you won't have extra charge.
You can also use WiFi.
You need to have an internet connection to use TextMe. It works on WiFi, 3G or 4G or Edge connections, whichever is available.