I've gotten the game to start the quest when I start a new game, but not when I load a saved game. To get to this point, I made the quest, checked the 'Start Game Enabled' box in the 'Quest Data' tab, then I created an index of 0 in the 'Quest Stages' tab. I checked the 'Start Up Stage' checkbox, and added a new, empty, Log Entry, to which I attached the papyrus fragment for my script.
Does anyone know what I'm missing to make the quest begin any time a game is loaded? Thanks for any help you can provide!

You could change the QUST's FormID with TESVSnip (just be sure to load/save w/ CK afterward) and it would be seen as a different form by TESV.exe. To wipe the quest each save load, use a GetGameLoaded() check and Reset() when true, hinging the rest of the code on 'Else'.