Only in it for the gold

Setting out. 

Imo the best motivation at the first is that they're only in it for the gold.

There's some disdain for the D&D cliche: meetup in a tavern, rumors of haunted ruins, great treasure, strap on yer boots and swords and set off to drag out the gold. 

To me, this is the best way (it doesn't have to be a tavern or exactly this way, but basically).  Why? Setting out on such a path - no patron hiring, no captive to rescue, no message from distant relatives, no master commanding, no mentor guiding, no specific mysteries to solve - demonstrates the trait that all these adventurers must share, appetite for risk.  They'd rather trudge off into the depths than follow tamer paths.

Now this is not to say that story and other motivations cannot develop.  They can.  In play. Possibly very soon after the start.  Including patrons, prisoners, masters, mentors, mysteries.  All that.  The second encounter in the dungeon might be with a group of warriors - men of a local noble - seeking their lord, the Duke, who is somewhere in the dungeon plumbing its depths for treasure (of course), and also looking for a rare plant apparently needed by a crazed holy man treating the Duchess who has contracted an unusual and mysterious contagion.  The holy man seems to have a strange powerful sway over the Duke.  The Duke's warriors don't trust the holy man.  They need to deliver a message to the Duke: his daughter has been kidnapped and hauled off somewhere within the dungeon. 

A prisoner, opportunity for a patron, mysteries - many possibilities are open to the PCs here. Will they involve themselves? What will they try to gain?  How? They could seek out the duke, offer their services, or hunt for the kidnappers, attempt to rescue the duke’s daughter, or try to steal her away and offer her for ransom themselves.  They could visit the holy man and the duchess, to learn about the disease and the plant, or try to discover what sway the holy man seems to have over the duke. Or ignore the whole thing.  Or something else I haven’t thought of.  They will devise schemes of their own in play.

The difference in my mind, between more extensive motivating backstories and my preference -  the tavern meetup/go get the gold scenario - is player choice.  There is one choice forced on the players at the outset in "my" way: their characters are not ultra-cautious types, they are willing to sign on for adventure.  Otherwise why not take up safer pursuits?  Follow mom or dad into a life of bartending, horsetraining, storekeeping, farming, whatever.  Possibly smarter paths.  But Game Over.  So their first choice is a forced one – the requirement that they are willing risk life and limb for the thrill and rewards of adventure, instead of heading for tamer pastures.  That is the only forced choice.  There’s no requirement at the start to hire on as bodyguards, save a village, rescue anybody, serve in a wizard’s employ, investigate mysterious disappearances, etc.  The choices whether to do these things come later in play.