I had to help someone do this today. I discovered I couldn't remember how to do it offhand and went and found my paper notes on it. That, by itself, told me that I needed to make this blog entry.
To start, save your game. Whenever you do something new and different, save the game before you begin.
Next, open the console. To do this with an English language keyboard hit the ~ key (usually found in the upper left hand corner of the keyboard). If you are NOT using an English language keyboard you can look here to find out how to open yours. While in the console you navigate using the PageUp and PageDown keys to scroll through the lists. You'll see in a minute why this is important.
With the console open type
and hit Enter. This should give you a list of all active quests. Finding the one you want is the trickiest part of this. The vanilla Skyrim quests tend to have easily recognizable names like CQ02 (for Companions Quest 2) or Favor154 (for an NPC favor). Usually mod authors will use a quest name that is reminiscent of the mod name or the name that they gave the quest and posted like FindArmor05 or efgFindArmor. When you've made your best guess as to the name of the quest that you want (and don't forget to navigate up the list with PageUp) move on to the next step.
For purposes of this blog entry I am going to assume the questID I'm looking for is FindArmor05. So, whenever you see FindArmor05 in this entry substitute the questID that you've chosen. So now type
and hit Enter. This will give you another long list. Each line will have two numbers on it. The one on the left is the quest stage and the one on the right is either a 0 or a 1. A 1 indicates that the stage to its left is complete and a 0 means that it has not yet been completed. Now find the first stage which has a 0 nest to it. For my purposes I'm going to assume that this is stage 50. So, when you follow the next instruction simply replace 50 with whatever stage you found for the quest.
setstage FindArmor05 50
and hit Enter. You should get a notice that a quest has been advanced. If it isn't the quest that you wanted that means you chose the wrong quest name from your list. Go back to your saved game and start over again ignoring that quest name this time. If, however, you chose the correct quest name then the stage that had been giving you trouble should be ended and you can simply close the console and move on with the game.
What if the stage that you bypassed involved collecting an item and it doesn't show up in the inventory so as to make progressing further impossible? No problem! Open the console again and type
help "Killian's armor"
and hit Enter where Killian's armor is the name of the item for which you are looking. You need the quotes if the item has more than one word in it. For a single word, like Wabbajack, the quotes are not needed. With a specific item you should only get one entry but sometimes you'll get more. A little common sense will get you by this. If one entry says (Armor) Killian's Armor 2b123456 and the other says (Book) Story of Killian's Armor 2b234567 I hope that you can readily decide to choose the armor (unless the book is the quest item you needed at this stage). At any rate, assuming the above numbers and that you're looking for the armor you will next type
player.additem 2b123456 1
and hit Enter. The number given at the end (1) is the number of versions of the item that you want. So, if the mod wanted an odd alchemical ingredient, say 5 of Killian's toenails which you've discovered (as above) to be 2b345678 you would type
player.additem 2b345678 5
and hit Enter.
This can more easily be done by using the Add Item Menu mod (if you already have it - do NOT add it just for this). Just open the AIM menu, choose the offending mod and select the item that you need.
And that should be all that's needed. For the record sqt is short for "show quest targets" and sqs is short for "show quest stages". In both cases just use the abbreviations as I posted them above.
Last note: it's kind of weird doing a blog post without pictures.