Full disclosure: I'm a newbie in this area. So while I've heard terms such as GSM and SMS I've never worked in this area. I do have smarter people to consult, though. ;)
In some of our industrial controller applications, we connect a wireless/cellular router to the AVR through a UART connection. We now apply only Cradlepoint IBR600LE, which I believe is 4G LTE? (we did also prototype with other models/brands that work equivalently)
I do the firmware for the AVR side. There is a provision for the "user" to connect (typically) a laptop through USB cable and FTDI chip to one of the AVR's USARTs. The AVR is always the "slave" in this setup, responding to requests from the PC "master". Typical uses are log file upload; configuration file upload/download; read status; and some "remote control" -- show app's display contents on the PC and simulate button presses back to the unit.
In some installations, we now connect to the Cradlepoint router. The functions can be then done from anywhere in the world. ;) From my firmware point of view, it is totally transparent -- plugging in the USB cable switches the USART to the USB; otherwise, the USART talks to the Cradlepoint.
As mentioned, my AVR is the slave and never speaks unless spoken to. The firmware doesn't "know" anything about the Cradlepoint -- no AT messages or the like. All configuration and such is done from the "other end".
Now, a need/desire has arisen to send unsolicited messages from the AVR app through the Cradlepoint to some remote destination. Alarm conditions; daily usage; like that.
[newbie alert] So, I think I want to send an unsolicited SMS message and have it show up e.g. as a text message on my cell phone? Or as an email message? Does anyone have experience to tell me where to get started?