When does the Alien Die reset?
The alien die resets between tasks.  The only exceptions are that you don't reset between mission tasks, and you don't reset between base defense tasks.

You reset between each science task and each global defense task, so you'll start each science card and each continent at 1.  For missions and base defense, you start the first one at 1, but from then on it will keep increasing until you move to the next type of task.