A major tip: Getting from the south of Boston (even a few miles out) to north of the city can be a pain at any time of day. If you are going to play Ipswich (north), make sure you stay either right in the city or just north of it. Granite Links is just south of the city (10 miles if that). While it may seem like a no-brainer travel wise, at the wrong time of day, it can take you 45 minutes from downtown Boston. Pinehills is in Plymouth, farther south, but not much worse of a drive (getting through the immediate city traffic is the hardest part).
As far as Ipswich, very nice course. I like it, but it is a little quirky. Fairly tight off the tee. Relax, club back, and pick your spots, you should be OK. Expect to lose a few balls - some holes, right off the fairway is New England never-never land (pine trees and a layer of leaves and undergrowth). Greens can be fast there, but my guess is in May they will still be getting up to speed. Nice clubhouse, everything is in a gated community (condos around the course). It is a nice take.
Greater Boston area (north, south, west) has some great private courses. If your local pro can call in a favor for you, it may be worth it. Between Ipswich and Boston, if you have a place in your heart for Donald Ross (and who doesn't?), see what connections you can work to get on at any of the following: Charles River, Braeburn, Winchester, Essex or Salem. Not coincidentally, not only are those Donald Ross courses, but those five are probably on anyone's top 10 courses in Massachusetts (and many might have them as their top five). Another public option, if you are staying north of the city is Bass Rocks in Gloucester (prounouced "glawstah"). It is pretty (on the ocean), short but public course.
In terms of public courses, there are some nice tracks, just not maintained, within the city limits. Franklin Park (aka. Devine) and George Wright are two great city courses (you will have to cab to them as driving in Boston is as hazardous as playing Pebble Beach in a typhoon, blindfolded, on one leg). No country club environment at those courses, but some history (Devine is a Ross layout), and a good reminder that golf cuts through the normal demographics that separate cities and suburbs. If anyone mentions Newton Commonwealth to you, un-friend them instantly. Once (circa 1930) a great golf course, over the years, it has been carved up and sold off for apartments and condos. It is a hard-hat-required course as I swear holes cross each other there. Brookline (very close to the city limits) has a public course (formerly Putterham Meadows, now called Lynch). It is a decent track that abuts The Country Club (if you are lucky enough to have in at TCC, by all means take it). If you ever read "Missing Links" by Rick Reilly, the juxtaposition of these two courses is the real life basis for the Mayflower and Ponkaquoque muni course (given the name, part of the basis too, undoubtedly was Ponkapoag - a 36 hole muni layout, not far from Boston, that is a mere shadow of its design. Ponky's architect? Donald Ross of course). Like Bostonians, golf in the city is more than meets the eye. Under a lot of rough exteriors lies some genuine charm.