Driving in downtown Boston is not particularly easy, and navigating is even harder. Parking is available here and there, particularly near common tourist attractions, but not overly plentiful and often rather expensive. Driving straight through underneath Boston is not too bad (outside of rush hours), but that's not too useful for visiting the city, and incidentally not available to RVs due to propane restrictions in the tunnels.
Like Don, I take mass transit when visiting Boston itself. I usually park at the Alewife station (the end of one of the subway lines) which has a large parking garage. During the week I gather it may fill up with commuters, but on the weekends there is always plentiful parking there, and it's pretty reasonable parking rates ($7 per day, I think). The riverside station also has a parking garage that I haven't used but that appears reasonably easy to get to and the MTBA website usually has parking availability during the week.
Pets are generally permitted on the subways and other mass transit during non-rush hours, though it is at the discretion of the operators. They are not allowed during rush hours.