On the OpenSprinkler circuit, one wire from 24VAC is tied to the circuit ground, the other is called the common (COM) wire. Each solenoid has one wire that’s connected to the common wire, the other tied to its corresponding triac’s T2 terminal. And you are right that the triac’s T1 terminal is connected to ground. When the triac is turned on (i.e. gate to ground voltage above threshold), the channel between T1 and T2 is open, thus completing the circuit (24VAC common -> solenoid -> triac -> 24VAC ground).
The resistor value is 300 ohm. But anywhere between 200 to 330 ohm should be fine. According to the triac’s datasheet, the gate to ground voltage drop is about 1 to 1.3V, and you need at least 5 to 7 mA of gate holding current for triac to remain open. So assuming the microcontroller outputs a voltage of Vcc (in the case of OpenSprinkler, Vcc = 3.3V), you can calculate the necessary resistor value. For example, (3.3V – 1V) / 7mA = 328.6 ohm.