Amtrak Northeast Regional operates a train from Providence to Groton every 3 hours. Tickets cost $70 - $190 and the journey takes 1 h 9 min. Alternatively, Greyhound USA operates a bus from Providence to Groton twice daily. Tickets cost $13 - $30 and the journey takes 1 h 41 min.
From Providence Amtrak train to New London Union Station Amtrak. 1 h 9 min $109. From Providence, RI bus to New London, CT. 1 h 41 min $23. From Providence drive to Groton. 59 min $6.