Steal ideas from the players. I had a game where the players had to acquire some magic rocks and bring them to the person that hired them. The rocks weren't important and were just a reason for them to go to a certain location. The players became convinced that the rocks were dragon eggs so I made them dragon eggs. Now they became very important as the PCs went to great lengths to hide these eggs from the bad guys.
Lots of good advice so far. For me the thing I'd add is to remember that unless a character in your game world wanted something to happen, it's irrelevant (this can be PCs or NPCs). Doesn't mean that stuff won't happen that's seemingly "random" or "just there for flavor"; just remember that those things aren't relevant unless they're tied to someone's motivations, either PC or NPC. The fastest, easiest way to make a game feel more like a "living world" is to have characters in that world have real motivations that are put into action.
The best stories aren't stories about stuff that happens in a vacuum; they're stories about people acting on the things that truly matter to them.
For me, the game world, the setting, the flavor, etc., can all take a back seat as long as there's people in that world doing interesting things that will have a meaningful impact on my character.