I usually like to figure out the over-arching themes that I want the campaign to focus on, and hard-wire them into the first adventure. In other words: the first adventure needs to set up what the campaign is going to be about.
If it's an exploratory hex crawl, the game starts with the PCs moving into the wilds in some way, past the bounds of civilization for the first time.
If it's about political machinations, the game starts with a dynamic introductions of factions and important NPCs at odds with each other.
If it's a classic adventure path towards a fight with the BBEG at the end, the first adventure is where the PCs learn something is wrong with the world.
Usually, I will also include a seemingly minor NPC, detail or plot point that will be revealed to be critically important much later in the campaign. Planting and payoff, accept no substitutes!
PCs will have preexisting relationships or be thrown together by circumstances and forced to work together (a patron, a disaster, a pressing need). Either can work, but I try to give PCs as many reasons to work together as early as possible so the players don't have to struggle to define new characters and new relationships all at once.