Logan airport is accessible on the T from both the Blue and the Silver lines. The silver line is a bus line that runs from the south end of Boston to the airport. The Blue line is part of the underground subway system in Boston. An easy connection to the Silver line is from the south station stop on the Red line. The Blue line is easily accessible from a number of underground stations in Boston or you can transfer to the Blue line from the Government Center spot.
Figure out what time you need to get to the airport and then go to mbta.com. Boston's public transportation MBTA site has a time calculator where you can input your starting and ending (Logan airport) address and will give you an estimated time. The site will also calculate the best route to take.
If you do not have access to a computer. get your hands on a mbta map. You can estimate your location to the closest T stop and then figure out out to get to the airport from there. The easiest way to get to the airport using the T is by going on the Blue line (if possible). The underground cars are faster and more reliable.
If you need to transfer to the Blue line from another line on the T, head toward Government Center. This is the central location of the T system and is very close to the airport stop. Once you are there hop on the Blue line headed outbound toward Wonderland. Then just wait until the airline stop.
Once you are at the Logan stop, catch a free shuttle bus that stops at each terminal at Logan.