There are a lot of options for that. As majenko said, you can use an Arduino + GSM Shield.
This will get you started.
... And if you want to be adventurous, then there are several options. You can use GSM modules for basic GSM connectivity. I recommend SIM900A or sim908 which also has a GPS built-in.
And a better option would be to use Quectel M66 GSM + Bluetooth module, because of the smaller form factor , class-12 GPRS (sim900 has class-10) and better power usage for longer battery life in case of battery operation.
These modules works on UART based AT commands. You can reference their respective Datasheets for that. To send AT commands you need a micro-controller with a built-in UART module.. there are a lot of options for this so I would let you research that.
But I would prefer atmega8 because it's cheap and easily available.