The quest cards have no quest number on the front. When are quest cards flipped over? When the setup or reaction text is fufilled or when a hero successfuly quests to that location?

The quest cards should be numbered in the format 1A, 1B, 2A, 2B, and so on. Usually the "A" side has initial setup instructions and fluff text. The "B" side usually has the number of quest progress tokens need and/or other requirements to move onto the next stage. I hope this helps.

Generally speaking, when you progress to a new quest card, you follow the instruction on the A side (though most of the side A of quest cards contains only flavor text), then flip it to side B immediately. You don't need to put progress token on side A in order to flip it to side B.

However, there is/are specific quest(s) that instruct you not to flip it to side B until certain timing (I only remember one such quest as in the Kazad-dum box). In that case, just follow this special rule.