I'd like a starting point to know in which protocol my very-low-level-phone, nokia 1110, is able to sens one of my sms to the short message service center of my provider.
My 'end goal' is RECEIVE an sms sent from my phone into a 'generic device' able to understand the protocol.
So. which protocol is used for real-world mobile phones?
In GSM networks, short messages are transported via Mobile Application Part(MAP), a part of the SS7 protocol family. In GSM, one design principle is to separate signalling and payload transport, i.e. signalling is done out-of-band. MAP is a signalling protocol, its main responsibility is setting up and tearing down payload communication channels. Contrary to the out-of-band principle, MAP transports short messages inline. This is actually the reason for the 140 char limit of short messages, it's the maximum the message can piggy-pack next to the protocol information.
MAP is an application layer protocol. Depending on the physical transmission channel (radio, copper, fiber, IP), MAP is transported over a variety of protocol stacks.
For implementing your goal "RECEIVE an sms sent from my phone into a 'generic device' ", you would need to decide where you want to connect your device to the mobile network. If you want to use the radio interface, that would require building your own so-called baseband chip.