Closure: your new best friend
In comic-book writing, "closure" is the technique of implying complex plot developments without actually showing them. In Snapchat, it's the technique of showing your viewers the lead-up before, and the results after, with none of the boring crap in the middle.
Let's say you're doing a Story about your trip from NYC to Boston, and it starts with a five-hour train ride. You might snap yourself boarding at Penn Station and deboarding in Back Bay, with a video of scenery passing by a moving train window between those two events. This way, viewers can infer that you've traveled to Boston, without having to actually watch you do so.
When it's over, say goodbye
When you upload Stories to Snapchat, each "piece" is added chronologically to the end, and each has its own 24-hour countdown timer. So depending on when someone watches your Story, they may not see the whole thing. So tell your viewers when one Story ends and another begins by using simple title cards, like the one above.