The South End is basically the gay social neighborhood of Boston, and it seems like a lot of guys there get gym memberships to Boston Sports Club (BSC). If you live in South End, you will be relatively close to BMC, since it's just down Mass Ave, and you can take the 1 bus if you need to (or ride a bike).
To get to Franciscan Hospital, you would take the Green Line (ugh), and get off at the Warren St stop. There seems to be several East Boston Neighrborhood Health Centers, but they're all relatively close to the Blue Line, and the one on Sumner St is right next to a T stop.
You could also check out bus routes that might be more direct.
South End is very central, has a lot of restaurants and galleries, and is a nice neighborhood - also kind of expensive though. Personally, I liked living in Central Sq up in Cambridge, but if I had to commute to three different locations like that, it might be a bit of a headache. Boston is small enough to get almost everywhere by bicycle, but drivers are idiots (you will get hit by a car if you ride enough), and the winter makes biking unpleasant at times.