You can pick for immediate use as soon as the skin is hard enough that you can't puncture it with your fingernail. At this point, most varieties are white-ish with a hint of green or yellow.

To store, they need to fully ripen on the vine, roughly 4 more weeks, in my experience, beyond the stage above, although longer is better on the vine if not in danger from pests, disease, or weather. By then, the basic S.S. is yellow-ish in color, as the yellow deepens as it ripens.

I was able to store my spaghetti squash for several months by keeping it in the cool of our basement storage room. I closed all heating vent to the room to keep it as cool as possible.