There are normally several active quests at a time available. The list here contains the quests on the left side of the game screen, sorted by topic and mostly in the sequence they appear.

Don't click on any box that pops up in the middle of the screen saying "Assemble"! Click the X and look for the quest associated with it or you might have to do the whole thing all over again. As long as you did not accept the concerned task before assembling the artifact you don't get credit for it!