Best and cheapest options are bus or train:
Train: NYAS express bus ($15) from terminal to Penn Station, then take Amtrak Northeast Regional train but be sure to take the train that stops at Hartford (NY-Springfield-Boston) as only some of the NE Regional trains take that route. Then local transportation to Manchester.
Bus: NYAS express bus from terminal to Port Authority Bus Terminal, then take Greyhound bus to Hartford.
Check Amtrak and Greyhound for schedules and fares. Taxi from terminal to either train station or bus terminal is flat rate $45+tolls+tip.
Best and cheapest options