There are no practical public transportation connections between Amherst and Bradley Hartford/Springfield International Airport (IATA Code: BDL). Your best bet is a private shuttle service called the Valley Transporter. Advance reservations are required and fare varies, but expect to pay about $53 each way for a shared van. The service recommends you plan on travel time of about 2 hours, but excepts no responsibility for missed flights due to delays.
Amherst has a small train station for Amtrak (train station code: AMM). The station has no services and a modest covered platform to wait for your train. Amtrak's "Vermonter" stops here once per day in each direction on its route between Washington, DC and St. Albans, Vermont. Be aware there is no way to purchase tickets in Amherst; your only option is to purchase tickets on Amtrak's website and have tickets mailed to you (it is currently not possible to print out your own online tickets -- only a receipt with which to print out your ticket at a "QuickTrak" machine -- and no such printing machine is available in Amherst).
The station is located on 13 Railroad Street, just south of Main Street and to the east of Amherst Center and north of Amherst College.
The primary means of public transportation to and from Amherst is the Peter Pan bus. This private regional bus line, founded in 1933, provides 7 daily buses between Amherst and Boston, normally with a connection in Springfield, total travel time between 3 and 3 1/2 hours. Occasionally on Fridays, there is additional express service to Boston. To get to Boston's Logan Airport, either transfer to a Peter Pan airport bus at Boston's bus terminal, or board the MBTA's "Silver Line" to the airport from nearby Boston South Station. It is also possible to connect in Springfield to buses to destinations such as Albany, Hartford, and New York City.
The main bus stop is in a bit of flux due to construction, but buses will normally depart from the town green across the street from the yellow information booth. Tickets can be purchased at Amherst Books on 8 Main Street, or online. Some buses also stop on the campuses of the University of Massachusetts and Hampshire College - check the website for details.