Connecticut is a state in the New England region of the northeastern United States. It is bordered by Rhode Island to the east, Massachusetts to the north, and the state of New York to the west and the south (with which it shares a water boundary in Long Island Sound). Popular cities in Connecticut include New Haven (the location of Yale University), Hartford (the capital of Connecticut and location of the Mark Twain house), and Mystic (home to the Mystic Seaport and Mystic Aquarium).
Metro North: http://mta.info You can take the train from Grand Central Terminal in New York to Union Station in New Haven. There are also branches that lead off the New Haven line to New Canaan, Danbury, and Waterbury. Best deal for time/price from NYC to New Haven.
Shore Line East: http://www.shorelineeast.com/ This train will take you from New Haven to New London.
Amtrak: http://www.amtrak.com Runs from California to Maine. Could take you from New haven to Hartford if necessary. Usually a more expensive option.
Megabus: http://us.megabus.com/ Offers trips between New Haven and NYC/Hartford/Boston for relatively cheap prices. Also can take their bus from Hartford to Storrs, Connecticut (home to the main branch of the University of Connecticut).
Greyhound: http://www.greyhound.com/ Could technically takes Greyhound buses from California to Connecticut. Has a few stops in Connecticut but sometimes pricier and more grittier than services like Megabus.