Hello!
I would suggest looking at Fairhaven and Dartmouth primarily. These towns are right on the interstate highway system (This area is about an hour from Boston) and share borders with New Bedford so commuting in/out of the city is easy.
You are right to be concerned about New Bedford's schools. They're terrible. The city itself has some wonderful historic districts and a great downtown area, but it has a large poor population which contributes to the troubles of the public school system (one of the highest dropout rates in MA). Fairhaven is a beautiful town sitting on New Bedford Harbor and Buzzards Bay. Run a quick google image search for Fairhaven High School... it's beautiful. As far as quality goes, Fairhaven is good. It's not the best, but it's pretty good and has good college placement and good test scores (keep in mind, MCAS scores are reflective of income as much as quality of school systems). You can find a 2 bedroom in Fairhaven for under $1,000 easily... perhaps even a 3 bedroom for that much. OH! and Fairhaven's music program is quite good. It's actually won awards. New Bedford next door has some very good extra-curricular music programs she could join as well as a very highly regarded
symphony orchestra that's involved in the community.
Dartmouth doesn't have the history and charm of Fairhaven (well, South Darmtouth does but it's very exclusive) but it has some good rental properties, suburban shopping (a mall and most chain box stores) as well as an excellent school system. Like Fairhaven, you can find rentals for under $1,000 in Dartmouth.
Mattapoisett and Marion are regarded as exclusive communities and to a degree, they are. However, there are good values to be found if you have patience. The schools in these towns (they actually share a high school) are up there with the best. Keep your eyes open because affordable rental properties in these towns go quickly. If you can find a place in one of these two towns, grab it as they're the best for schools in the area.
I'd also look into Freetown, Acushnet, and Lakeville. Possibly Rochester. These towns are a bit more rural and quiet which will make rentals harder to find but they rentals that do exist here are reasonably priced and schools are good (Freetown/Lakeville share a high school). I'd avoid Westport. While it's a beautiful town, the school district is sucky to put it nicely. Fall River is similar to New Bedford in that the schools are terrible. Avoid it. There are some great, very livable (and quite affordable) neighborhoods in New Bedford, but unless you plan on sending the kids to private school, I wouldn't do it... the schools are just that bad.
One option to consider for the short term is a winter rental. Towns like Fairhaven, Dartmouth, Marion and Mattapoisett are popular summer communities for tourists. What that means is that rental properties that sell for $1,000/week in the summer are usually open for cheap from September-May. These places are usually a great value for a short-term rental which isn't a bad idea if you're trying to get a feel for the area. You can probably snag one of these while you search for something a little more long-term.
Craigslist is a good place to begin looking for apartment. Check the "Southcoast" region of Massachusetts as that's technically where New Bedford is located. I'd also look at listings in the New Bedford
Standard Times Newspaper (link to their website... check the "marketplace" section.
THIS is the current apartment listings for the New Bedford area on South Coast House and Home. I noticed a lot of Middleboro apartments. This will make for a lengthy commute to New Bedford (30-40 minutes) but WILL give you access to Commuter Rail service to Boston from Middleboro. New Bedford will have rail service to Boston within 6 years, but for the short term, Middleboro is the closest station. Middleboro is geographically closer to Boston too, so you may be interested in that. I consider the commute to be a bit too much of a hassle, but it's not awful. Finally, the Fall River
Herald News may have some decent listings.
It's a nice area and one of the better kept secrets in Massachusetts which keeps rents lower. New Bedford is regarded as one of the most up and coming cities in the region. It still has a ways to go, but the revitalization, particularly in the downtown area (which has a huge chunk of it designated as a
National Historic Park), is absolutely stunning. Great restaurants, museums, shops, and entertainment. You're also an hour from Boston, 45 minutes from Cape Cod (though beaches in this area are in many cases better than Cape Cod beaches), 25 minutes from Providence and 45 minutes from Newport RI. New York City is a 3.5 hour drive. If you can locate near but not in New Bedford, you'll probably enjoy the relative value of it when compared to the rest of Massachusetts.
Any more specific questions, don't hesitate to ask. I'd also ask the moderator to move this thread out of the "Boston" subforum and into the Massachusetts one as even though New Bedford is close enough to Boston, it's not considered part of the metro area. You'll find that people in Massachusetts have an issue with distances... 1/2 hour is a LONG drive to many here.
Anyway, good luck!