Hi Dee,
I'm a native long islander (sorry) and spent many years in the DC metro area. We just relocated to NH from there 6 months ago. My kids are 9 and almost 5. A lot of my family is still on LI and in NJ so I am there frequently.
Base on how you described what you are looking for, my $0.02 would be to head to the Seacoast in some of the towns folks mentioned, and add in Durham (college town) and whatever else folks come up with here. Coming from areas where we come from, Portsmouth is, I think, one of the best choices to be in or near (depending on their schools, which I know nothing about.)
I was
very skeptical about no public water and the lack of trash pickup. After 6 months I can say that, for me, it's not at all a big deal - I had waaaay overblown that in my head. You can pay someone $5/week to do your trash if you want but we just take ours to the transfer station ourselves. The biggest changes for me in going from public water to a well are 1) use ridix once a month and 2) GET A GENERATOR because when you lose power you will eventually lose your water (no toilet flushing, no tap water, no showers.)
Taxes - it's all relative. Almost anywhere you go it will appear cheap to you. You also don't need to spend $500k on a house, though you well could. People say all the time that where I live (Sunapee area) is horrendously expensive. I suppose compared to most palces in NH it is, but compared to where I came from it's a total bargain for what you get. Just make sure you don't overpay for the
area you are in.
Traffic is another thing that will likely roll off you easily -- people complain about traffic on 93 all the time...to me, traffic is bumber to bumber on the Staten Island Expressway or the Belt at 1pm on a Thursday for no apparent reason. An extra 20 minutes to get to Boston during rush hour? Yeah - whatever.
WHat I did find was that when I go BACK to metro areas I'm near clasutrophobic! Not really, but you know what I mean -- I *really* notice the traffic, the speed at which life moves, how uptight people are, etc. I constantly feel like I am on vacation up here in NH.
If you like the outdoors and can find ways to enbrace winter activities I think you have a really excellent chance of loving it up here.
Good luck - Jackie
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What in particular brings you to New Hampshire? As you are by now realizing from the answers posted so far, finding public water & sewer, great schools, trash p/u, etc. may be a difficult mix, but not impossible. 
