You *can* use public transportation from Logan to the hotel. But I always took cabs. It's like 30 bucks (or so) and it saves the aggravation of dealing with it right when you get off a plane and just want to rest. You'd have to take a shuttle to the Blue Line airport stop from the airport. From there, you'd take the blue line to the State street exit, I think. I'm not entirely familiar with where your hotel is but you'd need to ask them what the closest T stop is. It may not be on the blue line, in which case you'd have to find a transfer point and transfer to another line to get to yours.
That may sound confusing, but I feel Boston's subway system is easily navigated once you get the hang of it. There's "outbound" trains that lead to all points away from the city, and "inbound" that lead toward it. Most people are actually quite friendly and will help you if you ask them directions, despite the stereotypes.
As for what to do... I, myself, didn't care at all for Quincy Market. I've got great Farmer's Market's in the town I live in here in California, and they don't smell like rotting fruit mixed with old fish. I had to cover my mouth and nose any time I went by it. Yuck.
There is of course, the famous Freedom Trail. You can rent an audio guide and follow it throughout Boston. My family was too lazy for that, hehe. We decided to do the Duck Tour instead. Most people I knew that live in Boston have never been on the Duck Tour, and think it's pretty lame. I went in not expecting much, but we all had the time of our lives. The drivers are a bit on the silly side, but you learn sooo much about the local past-and-present history... plus you get a neat little tour of the Charles River to boot!
Do NOT do the Beantown Trolley. This was an utter bore and rip-off, at least in our experience. We spent 45 minutes waiting for one to arrive and when it did it was packed to the brim and we were told to wait for the next one. When the next one arrived, we got on and sat down on the extremely uncomfortable wooden bench seats and were then stuck in traffic for a half hour. Granted, this could happen on the Duck Tour (make sure you don't tour at rush hour) but the guide was uninformed and boring, and stopped a couple times for smoke breaks! Ugh.
I agree that the New England Aquarium is a great place to go if you're into aquariums. I love the sea turtles.
Ironically there's a "Legal Seafood" restaurant right next door. Makes you wonder.
If you're into museums, there are a ton of them everywhere you look. My travel companions really wanted to see the glass flower exhibit at the Harvard Museum of Natural History, but we ran out of time. Harvard is easily accessible via the Red Line of the T.
As for jogging - you can do no better than the Esplanade - a park area along the banks of the Charles River. Everyone who's anyone jogs along that stretch. Just head toward the river and you'll find it. (I believe the Charles/MGH stop on the Red Line would plant you right there.)
I didn't often eat in Boston proper so I can't recommend much in the way of food. Jtammy has a good idea about the North End, though. Supposedly excellent italian food.
*shew*
Anything else I can help with?
Boston!
