Boy! That is really hard to say...what i would do is get the $25 parking pass at any of the Ranger stations (easy to find online). I would ask them for suggestions, they're very knowledgeable and extremely helpful. My favorite easy hike is Dianah's Bath, 30 minutes in, a fairly long "stair step" of falls with interesting geological features. A little crowded but worth it. Grest picnic area (no tables, no fires, adjust accordingly) Go beyond the falls for gentle climbs and far less crowding. Next, talk to everyone that you see, people who spend time in the woods take the time to talk to others you'll be amazed at the friendliness of 90% of the people up here. The people from up-tight cities even get more open/talkative up here (Nature does that to people).. You will get lots of useful information that way. All in all, bring a camera andgo for it, you and yours will have fond memories for a long time. ENJOY!
There's are a few hiking apps that detail many trails and use a rating system similar to ski slopes, green circle for easy trails and black diamond for difficult ones. REI has the hiking project app which I like. Mt Moosilauke is great via the old carriage road trail. Very easy with fantastic views from its exposed summit. Enjoy!
Tuckerman's ravine is a nice hike and there is also mt jasper it is not a long hike but great for kids and for a just a lite hike day.
Depending on your hiking experience Mount Washington via the Amonoosuc Trail. If you don't want to hike it you could take the Cog Railway Train or drive the auto road to the summit.
I would absolutely recommend the kangamangus Highway. There are many senic stops along the way as well as hiking trails. On either end of the highway is a busy tourist town full of shopping and fun things to do with the kids. If you love the outdoors there are many things to see in Franconia notch. The Basin being one of my favorites. If you are more interested is retail therapy there is Settlers Green in Conway. If you have young children i would also suggest that you check out Santa's Village and Story Land.
The Flume. Gotta go. Beautiful
The Basin is cool and you can hike up Cascade Brook Trail to Lonesome Lake. 4.5 ish mile round trip with 1000ft of elevation gain and some cool water crossings.
Arthrusea falls is a great hike and can be done with younger children. if looking for extreme id say mt Washington .
Hike or drive to top of Mt Washington Easy - artists bluff Medium - Zeacliff (best view in the whites!) Hard - Mt Lafayette
Thanks! Your answer is awaiting moderation.