If you are interested in American history, spend a day or two on the North Shore. Do the Salem Witch museum/walking tour. Stop at the Peabody Essex Museum, which has an amazing collection and is the oldest, continuously operating museum in the US. Tour the House of the Seven Gables, which was the inspiration for Nathaniel Hawthorne’s novel of the same name. When you are ready to eat, go to nearby Essex for fried clams at Woodman’s.
If you want to see a New England beach, you don’t have to go to Martha’s Vineyard. I would see the Cape/islands on another trip. Instead, drive up the road from Salem to Crane’s Beach, which is part of the Crane Estate in Ipswich. It’s a gorgeous beach. There’s also a large conservation area with trails nearby and Castle Hill (historic home)… Wingaersheek Beach in Gloucester (a town farther north) is also a beautiful beach. You can also go on whale watching excursions from Gloucester. Have a great trip.
From Cambridge you have easy access to a lot to see along the Charles. Downtown Boston is very walkable and if you follow the Freedom Trail from end to end, you would pass through many of the major areas to see. That’s a pretty long walk!
Without a car, making it to the Cape and then Martha’s Vineyard would be very tough to do in one day. Not even sure if that would be possible. Even with a car, getting to the ferry would take at least two and a half hours (but the traffic on the bridges might be bad anytime in August). Then there’s the ferry…and you’d want to hang on the island for a while. So scratch that unless you want to give it two days.
If you really want to head to the south shore, there are buses to Plymouth and Plimoth Plantation (living history museum) can be fun. The north shore definitely has more interesting towns to see (Gloucester, Salem). Any of those old fishing villages on the north shore are lovely with lots to see (and quite upscale).
The major neighborhoods in downtown Boston all have a unique flavor and are worth visiting. Back Bay, North End, Quincy Market/Faneuil Hall, Copley Square, Fenway. Boston Commons, Beacon Hill…all nice to walk around in. There will be plenty of restaurants pretty much everywhere.
The T is not the greatest rail system in the world, but it will be reasonably accessible to most places you want to see. There are commuter lines out of both and south station are quite extensive (you’d take one of those to Salem).
Enjoy. And you’ll get used to the local’s accent quickly.
Didn’t read carefully—without a car, your options are mostly limited to Cambridge and Boston, which still leaves lots to do. You will spend too much time traveling if you go far out of the Boston/Cambridge area. The commuter rail system really caters to commuters and options are limited during non-commuter times. You would have to take a bus to get to Falmouth to catch the ferry to MV or a bus to Hyannis to get to Nantucket. Neither is a day trip.
I would stick to sights in the Boston/Cambridge area—definitely do the Freedom Trail. Take one of the tours. If you like seafood, there are great restaurants in the Boston/Cambridge area. Lots of good suggestions for sightseeing above. One off-the-beaten track suggestion: take a walk in the Mount Auburn Cemetery in Cambridge—beautiful gardens and plantings and a favorite with bird watchers. It was one of the first garden-style cemeteries in the US designed as a park so people living in the city could get into nature.
"You must get a cannoli from Mikes, while you are there. "
Nope, Modern Pastry.
Or divide up and get cannolis from both and taste test. Cannoli wars. 
If you really want to go to Martha’s Vineyard and don’t mind paying some extra $$, fly in.
I recall a trip 30+ years ago where we enjoyed going to visit USS Constitution. I think we took a bike ride to get there(?)
There is also a ferry that runs from Boston Harbor to Salem and lands within about a 15 minute walk of the Peabody Essex Museum, with much of downtown within 10 minutes. It is likely quite a bit pricier than the commuter rail, but gives a chance to be on the water for about an hour if that appeals. Enjoy your trip. So much to see and do. We like walking the North End of Boston. Great restaurants, interesting food shops and near the Aquarium and waterfront.
Plimoth! Great idea. Worth renting a car to get out there. And the whale watching is nearby. Every single trip, we’ve seen plenty of whales (unlike my experience in S. CA.) Plus, it’s informative.
Thank you so much for all of your wonderful suggestions. It looks like it might be worth it to buy passes that admit us to multiple museums and attractions.