Re: Trip with kids and where to stay
Welcome to the forum. Are you looking to stay at a no frills campground where you won't be too far from various trailheads? Or staying at motels, cabins, etc? Or not so no frills camgrounds?
Except for once, we've always stayed at the simpler campgrounds. They run anywhere from $18 -$25 per night. If you head up from Boston on 93 (which we usually take because we live in Lawrence) and get off at exit 32 in Lincoln, you can head up the Kancamaugus Highway and they are located at various places. Closer to Lincoln, you could go to Bald Mtn/Artist Bluff (near Cannon). that was our first hike many years ago, a good one for kids. There is also a big campground, Lafayette, which is very popular. If you stay there, the trail to Lonesome Lake is right in the campground.
Rt 302 - Crawford Notch. There is the Zealand Campground and further up Zealand Road there are Sugarloaf I and II. Then you'll be near the Sugarloafs, which have great ledges and views. Go further up Zealand Road and you will come to the big parking area where you can take a 3 mile relatively flat but beautiful hike to Zealand Falls, where there is an AMC Hut (snacks, drinks, etc. and you can refill your bottles! I know you said they like climbing, after the hut it gets steep and you can get to the Zeacliffs, where there is a great view of Zealand Notch. Right off 302, near the Highland Center, there is Mt. Willard and many other possibilities.
North Conway area - we've stayed at Blackberry Crossing, one of those campgrounds along that end of the Kanc. Then you can look up lots of great possibilities - Boulder Loop, Moats (make it as long or short as you think they can handle), Middle and Peaked Mts (near Cranmore). Black Cap (near Cranmore). North Kearsage is nice - a restored fire tower up top.
I'm just scratching the surface here. My daughter Prema and I stay at these inexpensive places if we overnight (unless we have AMC Luxury Night, about once every year or two

).
I'm sure everyone else will soon jump in and give you some great ideas; one member has young kids and goes great places, another guy will sing the glories of Chocorua for you (lots of possibilities there, and it's south of the Conways).
Whatever you do, DO NOT FORGET THE BUG JUICE!