This is all demonstrated in The Sword, which you can download for free and that I linked to above. Here it is:
Burning Wheel The Sword Demo Adventure PDF
I'm not really inclined to write up a summary. You can look at the PCs and see their Beliefs (I quoted some salient ones upthread). You can see the setup (I quoted this fully, upthread). The scenario also gives some suggestions for how to frame a new scene/situation, based on the outcome. (I quoted some of this upthread.) The way it turns into a longer term campaign is to keep doing it.
Most of the obstacle in this scenario will be opposed checks (versus tests in the terminology of the game for which the scenario is written, ie Burning Wheel). But suppose someone tries to break the sword - then the obstacle would be taken from the list of obstacles in the rulebook. As I've already said, there are hundreds of sample obstacles provided.