MIAMI—Commuters and tourists fed up with South Florida’s clogged highways will soon have a new alternative: A posh passenger-rail service to whisk riders from Miami to Fort Lauderdale, West Palm Beach and, eventually, Orlando.

The $3 billion system, named All Aboard Florida but better known as Brightline, is privately funded and hasn’t received federal or state grant money. It comes online as President Donald Trump and Congress weigh a $1 trillion infrastructure plan that would rely heavily on private capital—leading some...