No problem MakeMeOT, moving to a new city is definitely tricky, but you're right it really is just about being smart. I don't know anything about Gerrish Street specifically, but I can tell you that there aren't really any t stops in Chelsea, so you'd probably have to take the bus if you weren't planning on driving.
tnsnc, that's kind of a hard question. The cost of living in Boston is kind of high, like most cities, but there are some areas that aren't bad price wise. I would recommend looking for a roommate though, if possible. I would also say to look along the orange and green lines of the t, so with price in mind, the Fenway/Kenmore area, Allston, Brighton, and the Back Bay for starters. The Fenway/Kenmore area is pretty nice and its about 20 minutes on the t to get to north station, where you can either take the Partners shuttle to MGH or you can walk (its about a 15 minute walk) Allston and Brighton are further out on the green line, but both are decent pricewise, its just a longer commute. The Navy Yard is a nice area, and the commute would basically be nonexistent, but it might be hard to find a place within your price range. Brookline is another really nice area, but again, rent is usually higher. I hope this helps! I'm in no way an expert, but I grew up/live right outside of Boston and I'm looking at moving closer myself, so I'm definitely willing to help haha.