Is there a train from Boston or Providence to Burlington?
No. You could take the train from Boston to Springfield MA and get a train to Burlington VT (runs once a day) or you could take a bus (Greyhound or Megabus) from Boston to Burlington.
Best to rent a car. It's a 3.5 hr drive from Boston to Burlington, without traffic.
There is Megabus. Maybe costs about $60 more or less depending on demand