I've attached a map by $/bedroom (from
this guy), with a few cheaper regions highlighted.
Route 28: Cheap, but not terribly accessible by T or by bike to Harvard because of the line of hills that runs from SE to NW up the middle of Somerville. Obama lived on Winter Hill, just NW of Route 28, when he was at Harvard Law. There's a big South American population. A little south and east is Union Square, which is where you will shop for groceries at 6:55am on a weekday, at the Somerville Market Basket. There's a straight shot with no hills from Union Square to the NE side of the Harvard campus via Washington/Kirkland, though no bike lanes on that route. A few good playgrounds, but not a lot of nice, big parks, though I haven't checked out the Mystic River Reservation across the river to the north. Near the Somerville Main Library.
North of Davis Sq: Near the Brown School, which is a good elementary school. Near the Davis T stop, which will take you fairly close to the incomparable Cambridge Main Public Library just east of Harvard Square. There's an entire floor of the library just for kids. Bikable to Harvard. Close to the open, green spaces at Tufts, decently close to Danehy Park. Being near the red line makes day trips into Boston easier. I recommend looking here.
Watertown: The 71 and 73 bus lines run straight to the Harvard T stop. The bike path along the river runs to Harvard. I know several Harvard grad students who live in Watertown. Parts of Watertown are a little more gritty and treeless than West Somerville/Davis. It's close to the business school. Also decently close to Fresh Pond and Mt Auburn Cemetery, both nice green places. I don't think of Watertown as being that walkable.
Belmont and Arlington have better schools but are farther. Parts of Arlington and Belmont have more of a small-town feel. I recommend not looking any farther east than Somerville. Transfers on the T add a lot of commute time, as do bottlenecks at bridges over rivers. I don't know much about Allston/Brighton.
For groceries, just go to a Market Basket, either in Somerville or Chelsea. If you really need to, there's a Costco in Everett. There are small ethnic grocery stores all over the place. An H-Mart (Korean) is opening soon in Central Square, Cambridge.
There are little parks, playgrounds, and libraries all over the place.
For second-hand clothing, go to the Goodwill store in Davis. There's a large children's section downstairs. Parking is pretty bad in Davis, though, so take a hand cart and walk in if you can. If you need a lot of clothes and have a day and a car, check out the Goodwill mother-ship in Roxbury.
Driving and parking in most of the Boston area are a pain. Getting an annual permit for street parking, at least near your apartment, is usually cheap and straightforward. Since your oldest is 6, you might be able to get away with mostly walking.
The whole lead paint thing is a pain. Landlords technically don't have to abate for families unless they've tested and found lead paint, but they usually don't want to take the risk. They do discriminate, it is illegal, but MA law makes the whole issue a really impractical pain for landlords ($10's of thousands to eliminate really a low risk). Landlords who do abate often will advertise that fact and then charge more. Maybe keep your eye out for Craigslist rental listing photos with unpainted trim, since the trim is where the lead paint usually was.
Biking is doable in the winter, but be prepared to walk in very bad weather. Biking is getting more popular in the Boston area, which makes a lot of sense. It's a pain to drive, and on a lot of streets, the cars don't go much faster than the bikes, which makes it somewhat safer. I don't feel confident enough to bike with the kids in a trailer.
If you use a wide stroller, keep in mind that some people don't shovel their sidewalks very well.
Keep your eye out for MMM meetups! PM me if you want to meet up when you get here.