Let me provide a slightly more jaded/realistic view of Montreal and Quebec as a New Yorker who has lived here since 2005. I promise, I am not trying to be a spoil-sport. I know several Americans who have moved here and really love it and a bunch who don't. In the end, I really think it depends on just how American your expectations are -- I'll elaborate on this in a bit.
Firstly, you certainly have the right attitude for life in Quebec. If you dive into the Francophone culture of Quebec, you'll be a step ahead. The language politics in Quebec are maddening to put it mildly but if you're able to stay above the fray by accepting that French is or should be the one and only language of Quebec, you'll stay happy. Be prepared to ignore the random Francophone who chooses to lecture your visiting family member for speaking English, or the one who goes off on a rant in a Korean restaurant because the proprietor is struggling in French. Laugh to yourself when you are ignored when you try to use English to explain a complicated issue to a government employee (yes, all these things have happened to me.) Let me say that I have met very, very many wonderful people in Quebec (jerks are not the majority here as they are not generally anywhere) but the jerks you meet will usually be jerks about language.
I have no personal experience of the school system in Quebec but one thing you should probably know is that the French public school system has outrageously high drop-out rates -- much higher than the English public school system which your children will not be allowed to attend. Sounds like this is not so much a problem for you but I'd also point out that your children will not be allowed to attend even an English private school (it's complicated to explain why, but that's the deal.) As an aside, the "English" schools are actually bilingual, usually offering half of their classes in French and half in English.
The healthcare system is crumbling; there's no way around that fact. Over one-third of the people living in Montreal do not have a regular doctor. Chances are that you will not either. The reason for this is also complicated (and political) but, trust me, this is the case. Further, if you need a specialist (which are even in shorter supply than GPs), you will be referred out and, as someone else has said, you will wait a long time. True story: last year, I got fairly sick and it was decided that I needed a colonoscopy. I was given a number to call and when I did, the receptionist told me to call back in six months as they had "closed their appointment book." (The current waiting time for a colonoscopy in Quebec is three years, by the way.) When I went back to my GP (who has opted out of the provincial health plan and so I have pay him -- only way to get a GP in Montreal) he told me "Yeah, it's not right but if pay them $500, they'll get you in within the week.) That's when I reinstated my American medical insurance. One odd twist in the whole thing is that, because of the doctor shortage, the province is looking the other way when doctors start private clinics (further exacerbating the shortage) but private, general health insurance is illegal. Go figure. Overcrowded hospitals are regular occurrences here. In fact, last time I was at St. Mary's they had posters up boasting that they were averaging only 10-20% overcrowding. Last time I was at the emergency room in New York, I waited maybe 15 minutes to be seen. When I took my boyfriend to the emergency room in Montreal, it took 14 hours for him to see a doctor (I'm not exaggerating.) And, don't think the language thing goes away when you're sick. There are some bilingual hospitals that are mostly downtown and on the Westside. If you go to a Francophone hospital, it is likely that you won't even be understood by those treating you.
Taxes, as another poster pointed out, are extraordinarily high. It is truly a shock even coming from a "high-tax" state like New York. You may wonder what you are paying for once you see/experience the crumbling infrastructure (and I'm not talking "well-used" like what you may see in New York: overpasses on major downtown highways are collapsing, a sinkhole formed in a major downtown intersection two weeks ago and three years ago a woman dining in a restaurant was killed when a concrete balcony fell on her -- corrupt building inspections. Today's newspaper ponders whether Old Montreal is next to crumble.)
Food costs just as much if not more than in New York (and the sales tax: 15.5%, is applied to things that it would not be applied to back home: clothes are taxed, for instance.) Further, the standard of customer service is very different in Quebec as well. I know it sounds petty, but you will be actually shocked at just how long you will wait to pay for groceries here. Also, the concept of 24 hour anything is pretty much non-existent.
Banks are a surprise as well. Do not expect that, as in the U.S., that a bank must legally clear a check in one or two days. Here it is entirely at the discretion of the bank. In fact, do not expect that a bank must legally treat a deposit made to an ATM machine like any other deposit as in the U.S.. I made a cash deposit at a ScotiaBank ATM (because the line was 20 people long with one teller working) and it took a week and a half for the bank to credit my account. When I called to complain, I was told, basically, if I didn't like it I was welcome to find a new bank. Oh,
and banks are open from like 10 am to 3 pm, just like they used to be in America...in the '60s. ;-)
Generally everything is more expensive here than in the US. And, there is much less variety in stores here. There's one middle-range department store (the Bay) and two very high-end department stores and that's about it. Everyone winds up at Canadian Tire which is kind of what you might expect if KMart and your local hardware store had a baby.
Let me say here that I realize one reason you say you'd like to move to Quebec is that it is not as "consumerist" or "mercantile" as, say, New York but I'm trying to point out the things that you will encounter that will be very different. Quebec is, frankly, a much less prosperous place than the Northeast; it may also be that there is less of a gap between rich and poor here, I will admit. However, salaries are (much) lower while prices and taxes are (significantly) higher. Real estate is relatively inexpensive, so that's one bonus. Quebec is also much less individualistic than the U.S. and I don't know how to explain this but you will find it cropping up in weird ways.
So, that leads me to the major realization that I had while living in Quebec: I'm an American. This came as a surprise to me. Before I moved here, I, too, loved the French Language and I, too, loved travelling abroad and sampling other cultures (lived in France for a year and the Netherlands for a year and a half.) I always felt like I was a 'citizen of the world.' In fact, I left New York for a lot of the reasons you say you would like to leave. However, for me, after a six month to a year honeymoon when I was entranced with the differences, Quebec never really added up. After a while I realized it's because I have very American expectations about some things: I expect medical care to be immediately available, for example; I expect to get good value and service when I'm paying for something; and I bristle when I encounter heavy-handed officialdom. Lastly, on the language issue, (and as an American, when you hear "language" you think "race" -- it's as touchy as that topic is back home) I found that having a love and respect for a language/culture wore off very quickly when I didn't feel that sentiment was returned.
In any case, I realize that some of this makes me sound like a jerk or that I'm trying to rain on your parade. I promise you, I'm not. The cultural differences between the Northeastern U.S. and Quebec are vast and in some ways hidden and you will find yourself puzzled over the smallest things. In the end, if you decide that Quebec is the place for you and your family, I wish you the absolute best of luck.
As to neighborhoods, if you want a French neighborhood, you might look at Rosemont, especially near Masson. It's a working class neighborhood that is quickly gentrifying so it is both very French, with some nice shops/restaurants and close enough to the center city that you could be near its amenities (and bilingual hospitals.)
**Lastly, please investigate fully what it will take to qualify to practice in Quebec if you are going to continue your career. Psychologists are abundant in Quebec/Montreal (one reason is that there is such a shortage of Psychiatrists.) I looked it up quickly and you will have to 1) Have your education certified by the Quebec Order of Psychologists as being equivalent to that offered in Quebec; 2) Pass the dreaded government French Language test; and 3) Pass a professional ethics course.
Quebec MD shortage spurs online pleas - Ottawa - CBC News
Montreal is falling down - Macleans.ca